Hochschild cohomology ring for self-injective algebras of tree class E 6 . II
aa r X i v : . [ m a t h . K T ] O c t HOCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRASOF TREE CLASS E . II. MARIYA KACHALOVA
Abstract.
We describe the Hochschild cohomology ring for a family of self-injective algebrasof tree class E in terms of generators and relations. Together with the results of the previouspaper, this gives a complete description of the Hochschild cohomology ring for a self-injectivealgebras of tree class E . Contents
1. Introduction 12. Statement of the main results 23. Bimodule resolution 64. The additive structure of HH ∗ ( R ) 205. Generators of HH ∗ ( R ) 256. Ω-shifts of generators of the algebra HH ∗ ( R ) 317. Multiplications in HH ∗ ( R ) 145References 1461. Introduction
Consider a self-injective basic algebra of finite representation type over an algebraically closedfield. According to Riedtmann’s classification, the stable AR -quiver of such an algebra can bedescribed with the help of an associated tree, which must be congruent with one of the Dynkindiagrams A n , D n , E , E , or E (see [1]). The complete description of the Hochschild cohomologyring was obtained for an algebras of the types A n and D n , see [2–5] (type A n ) and [6–11] (type D n ). For one of the two self-injective algebras of tree class E Hochschild cohomology ring wasobtained in [12].In this paper we consider the second part of algebras of tree class E . Any algebra of theclass E is derived equivalent to the path algebra for some quiver with relations. Namely, let Q s ( s ∈ N ) is the following quiver: Then any algebra of the class E is derived equivalent to one of the two following algebras:1) R s = K [ Q s ] /I , where K is a field, and I is the ideal in the path algebra K [ Q s ] of thequiver Q s , generated bya) all the paths of length 5;b) the expressions of the form: α t +2 α t +1 α t − α t + s )+2 α t + s )+1 α t + s ) ,α t γ t − α t + s ) − , t ∈ [0 , s − \ { , s } ,α γ s − α s − , α s γ s − α s − . R ′ s = K [ Q s ] /I ′ , where K is a field, and I ′ is the ideal in the path algebra K [ Q s ] of thequiver Q , generated bya) all the paths of length 5;b) the expressions of the form: α t +2 α t +1 α t − α t + s )+2 α t + s )+1 α t + s ) ,α t γ t − α t + s ) − . Henceforth we will often omit indexes in arrows α i and γ i as long as subscripts are clear fromthe context.The present paper is dedicated to the study of Hochschild cohomology ring structure for algebra R ′ s . We obtain the ring structure in terms of generators and relations. In studies of the structureof cohomology ring we will construct the bimodule resolution of R ′ s , which could be seen as awhole result. 2. Statement of the main results
In what follows, we assume n = 6.Let HH t ( R ) is the t th group of the Hochschild cohomology ring of R with coefficients in R .Let ℓ be the aliquot, and r be the residue of division of t by 11, m be the aliquot of division of r by 2. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 3 Consider the case of s >
1. To describe Hochschild cohomology ring of algebra R ′ s we mustintroduce the following conditions on an arbitrary degree t :(1) r = 0, ℓ ( n + s ) + m ≡ s ), ℓ ... 2 or char K = 2;(2) r = 0, ℓ ( n + s ) + m ≡ s + 1(2 s ), ℓ ... 2, char K = 3;(3) r = 1, ℓ ( n + s ) + m ≡ s ), ℓ ... 2 or char K = 2;(4) r = 1, ℓ ( n + s ) + m ≡ s (2 s ), ℓ ... 2 or char K = 2;(5) r = 2, ℓ ( n + s ) + m ≡ s + 1(2 s ), ℓ ... 2 or char K = 2;(6) r = 3, ℓ ( n + s ) + m ≡ s );(7) r = 3, ℓ ( n + s ) + m ≡ s (2 s ), char K = 2;(8) r = 4, ℓ ( n + s ) + m ≡ s + 1(2 s ), char K = 2;(9) r = 4, ℓ ( n + s ) + m ≡ s (2 s ), ℓ ... 2, char K = 3;(10) r = 4, ℓ ( n + s ) + m ≡ s );(11) r = 5, ℓ ( n + s ) + m ≡ s ), ℓ ... 2, char K = 3;(12) r = 5, ℓ ( n + s ) + m ≡ s (2 s ), ℓ ... 2, char K = 3;(13) r = 6, ℓ ( n + s ) + m ≡ s ), char K = 2;(14) r = 6, ℓ ( n + s ) + m ≡ s ), ℓ ... 2, char K = 3;(15) r = 6, ℓ ( n + s ) + m ≡ s (2 s );(16) r = 7, ℓ ( n + s ) + m ≡ s ), char K = 2;(17) r = 7, ℓ ( n + s ) + m ≡ s (2 s );(18) r = 8, ℓ ( n + s ) + m ≡ s ), ℓ ... 2 or char K = 2;(19) r = 9, ℓ ( n + s ) + m ≡ s ), ℓ ... 2 or char K = 2;(20) r = 9, ℓ ( n + s ) + m ≡ s (2 s ), ℓ ... 2 or char K = 2;(21) r = 10, ℓ ( n + s ) + m ≡ s + 1(2 s ), ℓ ... 2 or char K = 2;(22) r = 10, ℓ ( n + s ) + m ≡ s ), ℓ ... 2, char K = 3.Let M = 2 s gcd( n + s, s ) , M = ( M , char K = 2 or M ... 4;22 M otherwise. Remark . We will prove in paragraph 3 that the minimal period of bimodule resolution of R ′ s is M .Let { t ,i , . . . , t α i ,i } be a set of all degrees t , that satisfy the conditions of item i from the abovelist, and such that 0 t j,i < M ( j = 1 , . . . , α i ). Consider the set X = [ i =1 n X ( i ) t j,i o α i j =1 ∪ { T } , and define a graduation of polynomial ring K [ X ] such thatdeg X ( i ) t j,i = t j,i for all i = 1 , . . . ,
22 and j = 1 , . . . , α i ;( ◦ ) deg T = M. MARIYA KACHALOVA
Remark . Hereafter we shall use simplified denotation X ( i ) instead of X ( i ) t j,i , since lower indexesare clear from context. Denotation . e X ( i ) = ( X ( i ) , deg e X ( i ) < deg T ; T X ( i ) , otherwise.Define a graduate K -algebra A = K [ X ] /I , where I is the ideal generated by homogeneouselements corresponding to the following relations. X (3) X (2) = X (3) X (3) = X (3) X (5) = X (3) X (7) = X (3) X (8) = 0; X (3) X (9) = X (3) X (10) = X (3) X (11) = X (3) X (12) = X (3) X (13) = 0; X (3) X (14) = X (3) X (16) = X (3) X (17) = X (3) X (19) = X (3) X (21) = X (3) X (21) = 0; X (3) X (1) = e X (3) , X (3) X (4) = e X (5) , X (3) X (6) = e X (10) ; X (3) X (15) = e X (17) , X (3) X (18) = e X (19) , X (3) X (20) = e X (21) ; X (4) X (6) = ( e X (8) , char K = 2 , , otherwise; (r1) X (6) X (6) = ( − s e X (14) , char K = 3 , , otherwise; (r2) X (4) X (15) = ( e X (16) , char K = 2 , , otherwise; (r3) X (6) X (18) = ( − s e X (2) , char K = 3 , , otherwise; (r4) X (15) X (20) = ( e X (8) , char K = 2 , , otherwise; (r5) X (18) X (18) = ( s e X (12) , char K = 3 , , otherwise; (r6) X (18) X (20) = ( s e X (14) , char K = 3 , , otherwise . (r7)Describe the rest relations as a tables (numbers (r1)–(r7) in tables cells are the number ofrelation that defines a multiplication of the following elements). OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 5 X (1) X (2) X (4) X (6) X (7) X (8) X (9) X (11) X (1) X (1) X (2) X (4) X (6) X (7) X (8) X (9) X (11) X (2) − X (10) X (4) − X (10) X (11) X (6) (2) 0 0 sX (17) X (7) X (8) X (9) X (11) X (12) X (13) X (14) X (15) X (16) X (18) X (20) X (22) X (1) X (12) X (13) X (14) X (15) X (16) X (18) X (20) X (22) X (2) X (14) − X (21) X (4) X (14) X (17) X (20) X (6) X (19) − X (20) sX (5) X (7) X (19) X (21) X (8) X (21) X (9) X (19) X (21) − X (22) sX (3) sX (5) X (11) X (21) sX (5) X (12) X (13) X (14) X (15) X (16) X (18) X (20) X (22) X (12) − X (2) − X (10) X (13) X (3) X (5) X (7) X (10) X (14) X (15) − X (4) X (6) (5) X (11) X (16) X (8) X (18) (6) (7) − sX (17) X (20) X (22) Theorem 1.
Let s > , R = R ′ s is algebra of the type E . Then the Hochschild cohomology ring HH ∗ ( R ) is isomorphic to A as a graded K -algebra. Consider the case of s = 1.Let us introduce the set X ′ = X ∪ n X (23)0 , X (24)0 o , char K = 3; X ∪ n X (24)0 o , char K = 3; MARIYA KACHALOVA and define a graduation of polynomial ring K [ X ′ ] such thatdeg X ( i ) t j,i = t j,i for all i = 1 , . . . ,
22 and j = 1 , . . . , α i ;deg T = M (similar to ( ◦ ));deg X (23)0 = deg X (24)0 = 0 . Define a graduate K -algebra A ′ = K [ X ′ ] /I ′ , where I ′ is the ideal generated by homogeneouselements corresponding to the relations described in the case of s >
1, and by the followingrelations: X (1) X (23) = e X (23) , t = 0;0 , otherwise; X (1) X (24) = e X (24) , t = 0; e X (2) , t > K = 3;0 , otherwise; X (9) X (24) = − e X (10) ; X (15) X (24) = e X (14) , char K = 3;0 , otherwise; X (22) X (24) = − e X (21) ; X ( j ) X ( i ) =0 , j ∈ [2 , \ { , , } , i ∈ { , } , where t denotes a degree of the element X (1) . Theorem 2.
Let s = 1 , R = R ′ is algebra of the type E . Then the Hochschild cohomology ring HH ∗ ( R ) is isomorphic to A ′ as a graded K -algebra.Remark . From the descriptions of rings HH ∗ ( R ) given in theorems 1 and 2 it implies, inparticular, that they are commutative.3. Bimodule resolution
We will construct the minimal projective bimodule resolution of the R in the following form: · · · −→ Q d −→ Q d −→ Q d −→ Q ε −→ R −→ R . Then R – R -bimodules can be considered as leftΛ-modules. Denotations . (1) Let e i , i ∈ Z ( n +2) s = { , , . . . , ( n + 2) s − } , be the idempotents of the algebra K [ Q s ],that correspond to the vertices of the quiver Q s .(2) Denote by P i,j = R ( e i ⊗ e j ) R = Λ( e i ⊗ e j ), i, j ∈ Z ( n +2) s . Note that the modules P i,j , formsthe full set of the (pairwise non-isomorphic by) indecomposable projective Λ-modules.(3) For a ∈ Z , t ∈ N we denote the smallest nonnegative deduction of a modulo t with ( a ) t (inparticular, 0 ( a ) t t − OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 7 Let R = R ′ s . We introduce an automorphism σ : R → R , which is mapping as follows: σ ( e i ) = e n + s )+ i ,σ ( γ i ) = γ i + n , ( i ) s = s − − γ i + n , ( i ) s < s − , σ ( α i ) = − α n + s )+ i , ( i ) = 0 , ( i ) s < s ; α n + s )+ i , ( i ) = 0 , ( i ) s > s ; − α n + s )+ i , ( i ) = 1; − α n + s )+ i , ( i ) = 2 , ( i ) s > s ; α n + s )+ i , ( i ) = 2 , ( i ) s < s. Define the helper functions f : Z × Z → Z and h : Z × Z → Z , which act in the following way: f ( x, y ) = , x = y ;0 , x = y, h ( x, y ) = , x ... 2 , x < y ;0 , x ... 2 , x < y ;1 , x ... 2 , x > y ;0 , x ... 2 , x > y. Introduce Q r ( r m be the aliquot of division of r by 2 for considered degree r . Wehave Q m = s − M r =0 Q ′ m,r , m n − ,Q m +1 = s − M r =0 Q ′ m +1 ,r , m n − , where Q ′ m,r = f ( m, M i =0 P r + m ) − h ( m, i, r ⊕ f ( m, M i =0 1 M j =0 P r + m + js + f ( m, s + f ( m, s )+2 − h ( m, i (4 s +1) , r + js )+1 ⊕ f ( m, M i =0 1 M j =0 P ( r + m + js + f ( m, s + f ( m, s + f ( m, s )+1+ h ( m, i (4 s +1) , r + js )+2 ⊕ f ( m, M i =0 P r + m +1) − h ( m, i, r +3 ; MARIYA KACHALOVA Q ′ m +1 ,r = − f ( m, M i =0 P r + m )+1+ h ( m, f ( m, si, r ⊕ M j =0 P r + m +1+ js ) − h ( m, − f ( m, , r + js )+1 ⊕ M j =0 P r + m +1+ js + f ( m, s ) − h ( m, f ( m, , r + js )+2 ⊕ − f ( m, M i =0 P r + m +1)+1+ h ( m, − f ( m, si, r +3 . Now we shall describe differentials d r for r
10. Since Q i are direct sums, their elements canbe concerned as column vectors, hence differentials can be described as matrixes (which are beingmultiplied by column vectors from the right). Now let us describe the matrixes of differentialscomponentwisely. Remark . Numeration of lines and columns always starts with zero.
Denotations . (1) Denote by w i → j the way that starts in i th vertex and ends in j th.(2) Fot j th column of differential matrix let j be the aliquot and i be the residue of divisionof j by s .Define the helper functions f : Z × Z → Z , f : Z × Z → Z and f : Z × Z → Z , which act inthe following way: f ( x, y ) = , x < y ;0 x > y, f ( x, y ) = , x < y ; − x > y, f ( x, y ) = , x = y ; − x = y. Description of the d d : Q → Q – is an (7 s × s ) matrix . If 0 j < s , then ( d ) ij = w j + m ) → j + m )+1 ⊗ e j , i = ( j ) s ; − e j + m )+1 ⊗ w j → j +1 , i = j + s ;0 otherwise.If 2 s j < s , then ( d ) ij = w j + m )+1 → j + m )+2 ⊗ e j +1 , i = j − s ; − e j + m )+2 ⊗ w j +1 → j +2 , i = j + s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 9 If 4 s j < s , then ( d ) ij = w j + m )+2 → j + m )+3 ⊗ e j +2 , i = j − s ; − e j + m )+3 ⊗ w j +2 → j +3 , i = ( j ) s + 5 s ;0 otherwise.If 6 s j < s , then ( d ) ij = − e j + m +1) ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m +1) ⊗ e j +3 , i = j − s ;0 otherwise. Description of the d d : Q → Q – is an (6 s × s ) matrix . If 0 j < s , then( d ) ij = w j + m )+1+ j → j + m )+3 ⊗ w j → j + j , i = j + 2 j s, j − w j + m + s )+1+ j → j + m )+3 ⊗ w j → j + s )+ j , i = j + (2 j + 1) s, j s j < s , then( d ) ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = j − s + 1 , j < s ; w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + s + 1) s , j > s ; w j + m + s )+2 → j + m + s +1)+2 ⊗ e j + s )+1 , i = j + s ; e j + m + s +1)+2 ⊗ w j + s )+1 → j + s +1)+1 , i = j + s + 1 , j < s ; e j + m + s +1)+2 ⊗ w j + s )+1 → j + s +1)+1 , i = ( j + s + 1) s + 2 s, j > s ; w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ; w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+1 → j +3 , i = ( j ) s + 6 s ;0 otherwise.If 3 s j < s , then( d ) ij = e j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , j < s ; e j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = j − s + 1 , j > s ; w j + m )+3 → j + m +1)+1 ⊗ e j + s )+2 , i = j + s ; w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = ( j ) s + 6 s ;0 otherwise. If 5 s j < s , then( d ) ij = w j + m +(1 − f ( i ,s − s +1)+1 → j + m +2) ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m +(1 − f ( i ,s − s +1)+2 → j + m +2) ⊗ w j +3 → j +(1 − f ( i ,s − s +1)+1 , i = ( j + 1) s + 2 s ; w j + m +1)+3 → j + m +2) ⊗ w j +3 → j +(1 − f ( i ,s − s +1)+2 , i = ( j + 1) s + 4 s ; w j + m +1) → j + m +2) ⊗ e j +3 , i = j + s ; e j + m +2) ⊗ w j +3 → j +1)+3 , i = ( j + 1) s + 6 s ;0 otherwise. Description of the d d : Q → Q – is an (8 s × s ) matrix . If 0 j < s , then( d ) ij = w j + m − → j + m )+2 ⊗ e j , i = ( j ) s ; − f ( j, s ) e j + m )+2 ⊗ w j → j +1 , i = j + s ; f ( j, s ) w j + m )+1 → j + m )+2 ⊗ w j → j + s )+2 , i = ( j + s ) s + 3 s ;0 otherwise.If 2 s j < s , then( d ) ij = − f ( j, s ) f ( i , s − e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s ; w j + m )+2 → j + m )+3 ⊗ e j +1 , i = j − s ; − w j + m + s )+1 → j + m )+3 ⊗ w j +1 → j +2 , i = j + s ;0 otherwise.If 4 s j < s , then( d ) ij = w j + m )+3 → j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s , j < s − j = 6 s − w j + m + s )+1 → j + m +1) ⊗ e j +2 , i = j − s ; − e j + m +1) ⊗ w j +2 → j +3 , i = ( j ) s + 5 s ;0 otherwise.If 6 s j < s , then( d ) ij = − w j + m )+3 → j + m +1)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s , j < s − j = 8 s − − e j + m +1)+1 ⊗ w j +3 → j + s +1)+2 , i = ( j + s + 1) s + 3 s ; w j + m +1) → j + m +1)+1 ⊗ e j +3 , i = ( j ) s + 5 s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 11 Description of the d d : Q → Q – is an (9 s × s ) matrix . If 0 j < s , then( d ) ij = w j + m + j s )+2 → j + m )+3+ j ⊗ e j , i = ( j ) s ; − w j + m + s )+2+2 j → j + m )+3+ j ⊗ w j → j +2 j , i = j + s + 3 j s ; e j + m )+3+ j ⊗ w j → j + j s )+1+ j , i = j + (2 + j ) s ; w j + m )+3 → j + m )+3+ j ⊗ w j → j + s )+1 , i = ( j + s ) s + 2 s ;0 otherwise.If 2 s j < s , then( d ) ij = w j + m )+3 → j + m + s +1)+1 ⊗ e j +1 , i = j ; w j + m +1) → j + m + s +1)+1 ⊗ w j +1 → j +2 , i = j + 2 s ; e j + m + s +1)+1 ⊗ w j +1 → j +3 , i = ( j + s ) s + 6 s ;0 otherwise.If 4 s j < s , then( d ) ij = − f ( j, s − e j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , j > s ; w j + m +1) → j + m +1)+ j − ⊗ e j +( j − s )+2 , i = ( j ) s + 4 s ; w j + m +1)+1 → j + m +1)+ j − ⊗ w j +( j − s )+2 → j +3 , i = j + 2 s ;0 otherwise.If 6 s j < s , then( d ) ij = f ( j, s − e j + m + s +1)+2 ⊗ w j +2 → j +1) , i = j − s + 1 , j > s ; w j + m +1) → j + m + s +1)+1+ j − ⊗ e j +(7 − j ) s )+2 , i = ( j ) s + 5 s ; w j + m + s +1)+1 → j + m + s +1)+1+ j − ⊗ w j +(7 − j ) s )+2 → j +3 , i = ( j + s ) s + 6 s ;0 otherwise.If 8 s j < s , then( d ) ij = f ( j, s − w j + m + f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; f ( j, s − e j + m +1)+3 ⊗ w j +3 → j + f ( j, s − s +1)+1 , i = ( j + 1) s + 2 s ; w j + m +1)+1 → j + m +1)+3 ⊗ e j +3 , i = j − s ; − w j + m + s +1)+1 → j + m +1)+3 ⊗ e j +3 , i = j − s ;0 otherwise. Description of the d d : Q → Q – is an (8 s × s ) matrix . If 0 j < s , then( d ) ij = w j + m − → j + m )+1 ⊗ e j , i = j, j < s ; − f ( j, s ) w j + m ) → j + m )+1 ⊗ e j , i = ( j ) s + s ; − e j + m )+1 ⊗ w j → j + s )+1 , i = ( j + s ) s + 2 s ; e j + m )+1 ⊗ w j → j +2 , i = j + (5 − f ( j, s )) s ;0 otherwise.If 2 s j < s , then( d ) ij = w j + m )+3 → j + m +1) ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − j = 4 s − − f ( i , s − f ( j, s ) e j + m +1) ⊗ w j +1 → j +1) , i = ( j + 1) s + s ; w j + m + s )+1 → j + m +1) ⊗ e j +1 , i = j ; − w j + m + s )+2 → j + m +1) ⊗ w j +1 → j +2 , i = j + (4 − f ( j, s )) s ;0 otherwise.If 4 s j < s , then( d ) ij = e j + m )+3 ⊗ w j +2 → j +1) , i = ( j + 1) s , j < s − j = 6 s − w j + m )+1 → j + m )+3 ⊗ e j +2 , i = j + (1 − f ( j, s )) s ; − w j + m + s )+2 → j + m )+3 ⊗ e j +2 , i = j + (2 − f ( j, s )) s ; − f ( j, s ) e j + m )+3 ⊗ w j +2 → j +3 , i = ( j ) s + 8 s ;0 otherwise.If 6 s j < s , then( d ) ij = − f ( j, s ) w j + m )+3 → j + m +1)+2 ⊗ w j +3 → j +1) ,i = ( j + 1) s , j < s − j = 8 s − f ( j, s ) w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j + s +1)+1 , i = ( j + s + 1) s + 2 s ; − f ( j, s ) e j + m +1)+2 ⊗ w j +3 → j + s +1)+2 ,i = ( j + 1) s + 7 s, j < s − j = 8 s − − f ( j, s ) e j + m +1)+2 ⊗ w j +3 → j + s +1)+2 , i = ( j + 1) s + 5 s, s − j < s − w j + m )+3 → j + m +1)+2 ⊗ e j +3 , i = ( j ) s + 8 s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 13 Description of the d d : Q → Q – is an (9 s × s ) matrix . If 0 j < s , then( d ) ij = f ( j , w j + m )+1+ j +2 f ( j , → j + m +1) ⊗ w j → j + j , i = j + 2 j s, j f ( j , w j + m + s )+1+ j +2 f ( j , → j + m +1) ⊗ w j → j + s )+ j ,i = j + (2 j + 1) s, j s j < s , then( d ) ij = − f ( j, s ) w j + m + s +1)+1 → j + m + s +1)+ j ⊗ w j + s (2 − j ))+1 → j +1) , i = ( j + s + 1) s ; w j + m +1) → j + m + s +1)+ j ⊗ e j + s (2 − j ))+1 , i = ( j ) s + 2 s ; − w j + m )+3 → j + m + s +1)+2 ⊗ w j +1 → j +2 , i = j + 2 s, j > s ; − e j + m + s +1)+2 ⊗ w j +1 → j +3 , i = j + 5 s, j > s ;0 otherwise.If 3 s j < s , then( d ) ij = − f ( j, s ) w j + m +1)+1 → j + m +1)+ j − ⊗ w j + s ( j − → j +1) , i = ( j + 1) s ; w j + m +1) → j + m +1)+ j − ⊗ e j + s ( j − , i = ( j ) s + 3 s ; − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+1 → j + s )+2 , i = j + s, j > s ; e j + m +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 2 s, j > s ;0 otherwise.If 5 s j < s , then( d ) ij = w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+2 , i = j − s ; f ( j, s ) e j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + s ;0 otherwise.If 7 s j < s , then( d ) ij = w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = j − s + 1; w j + m +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; − e j + m +1)+3 ⊗ w j +3 → j + s +1)+2 , i = j − s + 1; − e j + m +1)+3 ⊗ w j +3 → j +1)+2 , i = ( j + 1) s + 4 s ; w j + m + s +1)+2 → j + m +1)+3 ⊗ e j +3 , i = j − s ; − w j + m +1)+2 → j + m +1)+3 ⊗ e j +3 , i = j ;0 otherwise. If 8 s j < s , then( d ) ij = − e j + m +2) ⊗ w j +3 → j + s +1)+1 , i = ( j + s + 1) s + 2 s ; w j + m +1)+2 → j + m +2) ⊗ e j +3 , i = j − s ;0 otherwise. Description of the d d : Q → Q – is an (8 s × s ) matrix . If 0 j < s , then( d ) ij = w j + m ) → j + m )+2 ⊗ e j , i = ( j ) s ; − w j + m )+1 → j + m )+2 ⊗ w j → j +1 , i = j + ( j + 1) s ; − e j + m )+2 ⊗ w j → j + s )+1 , i = j + (4 − j ) s ; e j + m )+2 ⊗ w j → j +2 , i = j + 5 s ;0 otherwise.If 2 s j < s , then( d ) ij = w j + m )+1 → j + m )+3 ⊗ e j +1 , i = j + s ( j − − w j + m + s )+2 → j + m )+3 ⊗ e j +1 , i = j + s ( j − − w j + m )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + 3 s ; e j + m )+3 ⊗ w j +1 → j +3 , i = ( j ) s + 7 s ;0 otherwise.If 4 s j < s , then( d ) ij = e j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s , j > s ; w j + m )+2 → j + m +1) ⊗ e j +2 , i = j + s ; − w j + m )+3 → j + m +1) ⊗ w j +2 → j +3 , i = j + 2 s, j > s ; − f ( j, s ) e j + m +1) ⊗ w j +2 → j +3 , i = ( j ) s + 8 s ;0 otherwise.If 6 s j < s , then( d ) ij = − w j + m +1) → j + m +1)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s , j < s ; e j + m +1)+1 ⊗ w j +3 → j +1)+1 , i = ( j + 1) s + s, j < s − j = 8 s − e j + m +1)+1 ⊗ w j +3 → j +1)+1 , i = ( j + 1) s + 2 s, s − j < s − w j + m )+3 → j + m +1)+1 ⊗ e j +3 , i = j + s, j < s ; − f ( j, s ) w j + m +1) → j + m +1)+1 ⊗ e j +3 , i = ( j ) s + 8 s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 15 Description of the d d : Q → Q – is an (6 s × s ) matrix . If 0 j < s , then ( d ) ij = w j + m )+2 → j + m )+3 ⊗ e j , i = j ; − w j + m + s )+2 → j + m )+3 ⊗ e j , i = j + s ; e j + m )+3 ⊗ w j → j +1 , i = j + 2 s ; − e j + m )+3 ⊗ w j → j + s )+1 , i = j + 3 s ;0 otherwise.If s j < s , then( d ) ij = − e j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + s + 1) s ; w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+1 , i = j + s ; w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ; − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 5 s ;0 otherwise.If 3 s j < s , then( d ) ij = w j + m +1) → j + m +1)+1 ⊗ e j + s )+2 , i = j + s ; e j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = ( j ) s + 6 s ;0 otherwise.If 5 s j < s , then( d ) ij = w j + m +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j +1) , i = ( j + 1) s + s ; − w j + m +1)+3 → j + m +2) ⊗ w j +3 → j +1+(1+ f ( j, s − s )+1 , i = ( j + 1) s + 2 s ; − e j + m +2) ⊗ w j +3 → j +1+(1+ f ( j, s − s )+2 , i = ( j + 1) s + 4 s ; − e j + m +2) ⊗ w j +3 → j +1+ sf ( j, s − , i = ( j + 1) s + 5 s ; w j + m + s +1)+1 → j + m +2) ⊗ e j +3 , i = j + s ; w j + m +1)+1 → j + m +2) ⊗ e j +3 , i = j + 2 s ;0 otherwise. Description of the d d : Q → Q – is an (7 s × s ) matrix . If 0 j < s , then( d ) ij = w j + m − → j + m )+3 ⊗ e j , i = j ; − f ( j, s − e j + m )+3 ⊗ w j → j +1) , i = ( j + 1) s ; − w j + m )+2 → j + m )+3 ⊗ w j → j +1 , i = j + s ; w j + m + s )+2 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 2 s ; w j + m + s )+1 → j + m )+3 ⊗ w j → j +2 , i = j + 3 s ; − w j + m )+1 → j + m )+3 ⊗ w j → j + s )+2 , i = j + 4 s ;0 otherwise.If s j < s , then( d ) ij = w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +1) , i = ( j + 1) s , j < s − j = 3 s − w j + m + s )+2 → j + m +1) ⊗ e j + s )+1 , i = j ; − w j + m )+1 → j + m +1) ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ; e j + m +1) ⊗ w j + s )+1 → j +3 , i = ( j ) s + 5 s ;0 otherwise.If 3 s j < s , then( d ) ij = − w j + m )+3 → j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , j < s − j = 5 s − w j + m )+1 → j + m +1)+1 ⊗ e j + s )+2 , i = j ; − e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 , i = ( j + s + 1) s + 3 s ; − w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = ( j ) s + 5 s ;0 otherwise.If 5 s j < s , then( d ) ij = w j + m )+3 → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s , s − j < s − e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = ( j + s + 1) s + s ; w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = ( j + 1) s + 3 s ; w j + m +1) → j + m + s +1)+2 ⊗ e j +3 , i = ( j ) s + 5 s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 17 Description of the d d : Q → Q – is an (6 s × s ) matrix . If 0 j < s , then ( d ) ij = w j + m )+3 → j + m +1) ⊗ e j , i = j ; e j + m +1) ⊗ w j → j +1 , i = j + s ; − e j + m +1) ⊗ w j → j + s )+1 , i = j + 2 s ;0 otherwise.If s j < s , then( d ) ij = w j + m +1) → j + m +1)+1 ⊗ e j + s )+1 , i = j ; e j + m +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ;0 otherwise.If 3 s j < s , then( d ) ij = w j + m +1)+1 → j + m +1)+2 ⊗ e j + s )+2 , i = j ; e j + m +1)+2 ⊗ w j + s )+2 → j +3 , i = ( j ) s + 5 s ;0 otherwise.If 5 s j < s , then( d ) ij = f ( j, s − e j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m + s +1)+2 → j + m +1)+3 ⊗ e j +3 , i = j ; − w j + m +1)+2 → j + m +1)+3 ⊗ e j +3 , i = j + s ;0 otherwise. Description of the d d : Q → Q – is an (6 s × s ) matrix . If 0 j < s , then( d ) ij = − f ( j, s − e j + m +1) ⊗ w j → j +1) , i = ( j + 1) s ; − f ( j , w j + m + s )+1+ j → j + m +1) ⊗ w j → j +1+ j , i = j + (2 j + 1) s, j f ( j , w j + m )+ j → j + m +1) ⊗ w j → j + s )+ j , i = j + 2 j s, j If s j < s , then( d ) ij = f ( i , s − f ( j, s ) w j + m +1) → j + m +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m )+1 → j + m +1)+1 ⊗ e j + s )+1 , i = j ; − e j + m +1)+1 ⊗ w j + s )+1 → j + s +1)+1 , i = ( j + s + 1) s + s ; − w j + m )+2 → j + m +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ; − f ( j, s ) w j + m )+3 → j + m +1)+1 ⊗ w j + s )+1 → j +3 , i = ( j ) s + 5 s ;0 otherwise.If 3 s j < s , then( d ) ij = − f ( i , s − f ( j, s ) w j + m +1) → j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+1 , i = ( j + s + 1) s + s ; w j + m )+2 → j + m +1)+2 ⊗ e j + s )+2 , i = j ; − e j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 , i = ( j + s + 1) s + 3 s ; f ( j, s ) w j + m )+3 → j + m +1)+2 ⊗ w j + s )+2 → j +3 , i = ( j ) s + 5 s ;0 otherwise.If 5 s j < s , then( d ) ij = − f ( j, s − w j + m +1) → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m +1)+1 → j + m +1)+3 ⊗ w j +3 → j + s +1)+1 , i = j − s + 1; − w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1)+1 , i = ( j + 1) s + s ; − w j + m +1)+2 → j + m +1)+3 ⊗ w j +3 → j + s +1)+2 , i = j − s + 1; w j + m + s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1)+2 , i = ( j + 1) s + 3 s ; w j + m )+3 → j + m +1)+3 ⊗ e j +3 , i = j ; − f ( j, s − e j + m +1)+3 ⊗ w j +3 → j +1)+3 , i = ( j + 1) s + 5 s ;0 otherwise. Theorem 3.
Let R = R ′ s is algebra of the type E ′ . Then the minimal projective resolution ofthe Λ -module R is of the form: (+) · · · −→ Q d −→ Q d −→ Q d −→ Q ε −→ R −→ , where ε is the multiplication map ( ε ( a ⊗ b ) = ab ) ; Q r ( r and d r ( r were describedbefore; further Q ℓ + r , where ℓ ∈ N and r , is obtained from Q r by replacing every directsummand P i,j to P σ ℓ ( i ) ,j correspondingly ( here σ ( i ) = j , if σ ( e i ) = e j ) , and the differential d ℓ + r is obtained from d r by act of σ ℓ by all left tensor components of the corresponding matrix. To prove that the terms Q i are of this form we introduce P i = Re i is the projective cover ofthe simple R -modules S i , corresponding to the vertices of the quiver Q s . We will find projectiveresolutions of the simple R -modules S i . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 19 Denotation . For R -module M its m th syzygy is denoted by Ω m ( M ). Remark . From here we denote the multiplication homomorphism from the right by an element w by w . Lemma 4.
The begin of the minimal projective resolution of S r is of the form · · · −→ P r +3)+3 ( α − α ) −→ P r +3)+2 ⊕ P r + s +3)+2 ( α α ) −→−→ P r +3) ( γα γα ) −→ P r +2)+1 ⊕ P r + s +2)+1 ( αγ − α α ) −→−→ P r +1)+3 ⊕ P r +2) ( α γα − α ) −→ P r +1)+2 ⊕ P r + s +1)+2 ( α γ α γ ) −→−→ P r +3 ( α − α ) −→ P r +1 ⊕ P r + s )+1 ( α α ) −→ P r −→ S r −→ . At that Ω ( S r ) ≃ S r +4)+3 . Lemma 5.
The begin of the minimal projective resolution of S r +1 is of the form · · · −→ P r +2 α −→ P r +1 −→ S r +1 −→ . At that Ω ( S r +1 ) ≃ S r +1)+2 . Lemma 6.
The begin of the minimal projective resolution of S r +2 is of the form · · · −→ P r + s +4)+1 α −→ P r +4) γα −→ P r +3)+2 α γ −→ P r +2)+3 ( α − α ) −→−→ P r +2)+1 ⊕ P r + s +2)+2 ( α α ) −→ P r +2) γα −→−→ P r + s +1)+1 αγ −→ P r +3 α −→ P r +2 −→ S r +2 −→ . At that Ω ( S r +2 ) ≃ S r + s +5)+1 . Lemma 7.
The begin of the minimal projective resolution of S r +3 is of the form · · · −→ P r +1) γ −→ P r +3 −→ S r +3 −→ . At that Ω ( S r +3 ) ≃ S r +2) .Proof. Proofs of the lemmas consist of direct check that given sequences are exact, and it isimmediate. (cid:3)
We shall need the Happel’s lemma (see [13]), as revised in [3]:
Lemma 8 (Happel) . Let · · · → Q m → Q m − → · · · → Q → Q → R → be the minimal projective resolution of R . Then Q m ∼ = M i,j P dim Ext mR ( S j ,S i ) i,j . Proof of the theorem 3.
Descriptions for Q i immediately follows from lemmas 4 – 7 and Happel’slemma.As proved in [14], to prove that sequence (+) is exact in Q m ( m
11) it will be sufficientto show that d m d m +1 = 0. It is easy to verify this relation by a straightforward calculation ofmatrixes products.Since the sequence is exact in Q , it follows that Ω ( Λ R ) ≃ R σ , where Ω ( Λ R ) = Im d is the 11th syzygy of the module R , and R σ is a twisted bimodule. Hence, an exactness in Q t ( t >
11) holds. (cid:3)
We recall that for R -bimodule M the twisted bimodule is a linear space M , on which left actright acts of the algebra R (denoted by asterisk) are assigned by the following way: r ∗ m ∗ s = λ ( r ) · m · µ ( s ) for r, s ∈ R and m ∈ M, where λ, µ are some automorphisms of algebra R . Such twisted bimodule we shall denote by λ M µ . Corollary 9.
We have isomorphism Ω ( Λ R ) ≃ R σ . Proposition 10.
Automorphism σ has a finite order, and (1) if char K = 2 , then order of σ is equal to s gcd( n + s, s ) ; (2) if char K = 2 , then order of σ is equal to s gcd( n + s, s ) , if s gcd( n + s, s ) is divisible by , and to s gcd( n + s, s ) otherwise. Proposition 11.
The minimal period of bimodule resolution of R is
11 deg σ . The additive structure of HH ∗ ( R ) Proposition 12 (Dimensions of homomorphism groups, s > . Let s > and R = R ′ s is algebraof the type E . Next, t ∈ N ∪ { } , ℓ be the aliquot, and r be the residue of division of t by . (1) If r = 0 , then dim K Hom Λ ( Q deg , R ) = s, ℓ ( n + s ) + m ≡ s ) or ℓ ( n + s ) + m ≡ s );2 s, ℓ ( n + s ) + m ≡ s (2 s ) or ℓ ( n + s ) + m ≡ s + 1(2 s );0 , otherwise. (2) If r = 1 , then dim K Hom Λ ( Q deg , R ) = s, ℓ ( n + s ) + m ≡ s );5 s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (3) If r ∈ { , } , then dim K Hom Λ ( Q deg , R ) = s, ℓ ( n + s ) + m ≡ s ) or ℓ ( n + s ) + m ≡ s + 1(2 s ); s, ℓ ( n + s ) + m ≡ s (2 s ) or ℓ ( n + s ) + m ≡ s );0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 21 (4) If r ∈ { , , } , then dim K Hom Λ ( Q deg , R ) = s, ℓn + m ≡ s );0 , otherwise. (5) If r = 4 , then dim K Hom Λ ( Q deg , R ) = s, ℓ ( n + s ) + m ≡ s );6 s, ℓ ( n + s ) + m ≡ s (2 s );5 s, ℓ ( n + s ) + m ≡ s );7 s, ℓ ( n + s ) + m ≡ s + 1(2 s );0 , otherwise. (6) If r = 6 , then dim K Hom Λ ( Q deg , R ) = s, ℓ ( n + s ) + m ≡ s );5 s, ℓ ( n + s ) + m ≡ s (2 s );6 s, ℓ ( n + s ) + m ≡ s );2 s, ℓ ( n + s ) + m ≡ s + 1(2 s );0 , otherwise. (7) If r = 9 , then dim K Hom Λ ( Q deg , R ) = s, ℓ ( n + s ) + m ≡ s );7 s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (8) If r = 10 , then dim K Hom Λ ( Q deg , R ) = s, ℓ ( n + s ) + m ≡ s ) or ℓ ( n + s ) + m ≡ s );6 s, ℓ ( n + s ) + m ≡ s (2 s ) or ℓ ( n + s ) + m ≡ s + 1(2 s );0 , otherwise.Proof. The dimension dim K Hom Λ ( P i,j , R ) is equal to the number of linear independent nonzeropaths of the quiver Q s , leading from j th vertex to i th, and the proof is to consider cases r = 0, r = 1 etc. (cid:3) Proposition 13 (Dimensions of homomorphism groups, s = 1) . Let R = R ′ is algebra of thetype E . Next, t ∈ N ∪ { } , ℓ be the aliquot, and r be the residue of division of t by . (1) If r ∈ { , , , , } , then dim K Hom Λ ( Q deg , R ) = 8 . (2) If r = 1 , then dim K Hom Λ ( Q deg , R ) = , ℓ ... ℓ ... . (3) If r = 2 , then dim K Hom Λ ( Q deg , R ) = , ℓ ... ℓ ... . (4) If r = 4 , then dim K Hom Λ ( Q deg , R ) = , ℓ ... ℓ ... . (5) If r = 6 , then dim K Hom Λ ( Q deg , R ) = , ℓ ... ℓ ... . (6) If r = 8 , then dim K Hom Λ ( Q deg , R ) = , ℓ ... ℓ ... . (7) If r = 9 , then dim K Hom Λ ( Q deg , R ) = , ℓ ... ℓ ... . Proof.
The proof is basically the same as proof of proposition 12. (cid:3)
Proposition 14 (Dimensions of coboundaries groups) . Let R = R ′ s is algebra of the type E ,and let ( × ) 0 −→ Hom Λ ( Q , R ) δ −→ Hom Λ ( Q , R ) δ −→ Hom Λ ( Q , R ) δ −→ . . . be a complex, obtained from minimal projective resolution (+) of algebra R , by applying functor Hom Λ ( − , R ) .Consider coboundaries groups Im δ s of the complex ( × ) . Let ℓ be the aliquot, and r be theresidue of division of t by , m be the aliquot of division of r by . Then :(1) If r = 0 , dim K Im δ s = s − , ℓ ( n + s ) + m ≡ s ) , ℓ ... or char K = 2;6 s, ℓ ( n + s ) + m ≡ s ) , ℓ ... , char K = 2;2 s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (2) If r = 1 , dim K Im δ s = s, ℓ ( n + s ) + m ≡ s );3 s − , ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... or char K = 2;3 s, ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... , char K = 2;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 23 (3) If r = 2 , dim K Im δ s = s, ℓ ( n + s ) + m ≡ s ); s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (4) If r = 3 , dim K Im δ s = s − , ℓ ( n + s ) + m ≡ s );7 s − , ℓ ( n + s ) + m ≡ s (2 s ) , char K = 2;7 s, ℓ ( n + s ) + m ≡ s (2 s ) , char K = 2;0 , otherwise. (5) If r = 4 , dim K Im δ s = s, ℓ ( n + s ) + m ≡ s );6 s − , ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... , char K = 3;6 s, ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... or char K = 3;0 , otherwise. (6) If r = 5 , dim K Im δ s = s − , ℓ ( n + s ) + m ≡ s ) , ℓ ... , char K = 3;6 s, ℓ ( n + s ) + m ≡ s ) , ℓ ... or char K = 3;2 s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (7) If r = 6 , dim K Im δ s = s − , ℓ ( n + s ) + m ≡ s ) , char K = 2;7 s, ℓ ( n + s ) + m ≡ s ) , char K = 2;5 s − , ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (8) If r = 7 , dim K Im δ s = s, ℓ ( n + s ) + m ≡ s );3 s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (9) If r = 8 , dim K Im δ s = s − , ℓ ( n + s ) + m ≡ s ) , ℓ ... or char K = 2;3 s, ℓ ( n + s ) + m ≡ s ) , ℓ ... , char K = 2; s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise. (10) If r = 9 , dim K Im δ s = s, ℓ ( n + s ) + m ≡ s );6 s − , ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... or char K = 2;6 s, ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... , char K = 2;0 , otherwise. (11) If r = 10 , dim K Im δ s = s − , ℓ ( n + s ) + m ≡ s ) , ℓ ... , char K = 3;2 s, ℓ ( n + s ) + m ≡ s ) , ℓ ... or char K = 3;6 s, ℓ ( n + s ) + m ≡ s (2 s );0 , otherwise.Proof. The proof is technical and consists in constructing the image matrixes from the descriptionof differential matrixes and the subsequent computations of the ranks of image matrixes. (cid:3)
Theorem 15 (Additive structure, s > . Let s > and R = R ′ s is algebra of the type E . Next, t ∈ N ∪ { } , ℓ be the aliquot, and r be the residue of division of t by , m be the aliquot ofdivision of r by . Then dim K HH t ( R ) = 1 , if one of the following conditions takes place :(1) r ∈ { , , , } , ℓ ( n + s ) + m ≡ s ) , ℓ ... or char K = 2 ; (2) r = 0 , ℓ ( n + s ) + m ≡ s + 1(2 s ) , ℓ ... , char K = 3 ; (3) r ∈ { , } , ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... or char K = 2 ; (4) r ∈ { , } , ℓ ( n + s ) + m ≡ s + 1(2 s ) , ℓ ... or char K = 2 ; (5) r = 3 , ℓ ( n + s ) + m ≡ s ) ; (6) r = 3 , ℓ ( n + s ) + m ≡ s (2 s ) , char K = 2 ; (7) r ∈ { , } , ℓ ( n + s ) + m ≡ s (2 s ) , ℓ ... , char K = 3 ; (8) r = 4 , ℓ ( n + s ) + m ≡ s ) ; (9) r = 4 , ℓ ( n + s ) + m ≡ s + 1(2 s ) , char K = 2 ; (10) r = 5 , ℓ ( n + s ) + m ≡ s ) , ℓ ... , char K = 3 ; (11) r ∈ { , } , ℓ ( n + s ) + m ≡ s ) , char K = 2 ; (12) r ∈ { , } , ℓ ( n + s ) + m ≡ s (2 s ) ; (13) r = 6 , ℓ ( n + s ) + m ≡ s ) , ℓ ... , char K = 3 ; (14) r = 10 , ℓ ( n + s ) + m ≡ s ) , ℓ ... , char K = 3 .In other cases dim K HH t ( R ) = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 25 Proof.
As dim K HH t ( R ) = dim K Ker δ t − dim K Im δ t − , and dim K Ker δ t = dim K Hom Λ ( Q t , R ) − dim K Im δ t , the assertions of theorem easily follows from propositions 12 – 14. (cid:3) Theorem 16 (Additive structure, s = 1) . Let R = R ′ is algebra of the type E . Next, t ∈ N ∪{ } , ℓ be the aliquot, and r be the residue of division of t by . ( a ) dim K HH t ( R ) = 3 , if t = 0 . ( b ) dim K HH t ( R ) = 2 , if one of the following conditions takes place :(1) r ∈ { , } , t > , ℓ + m ... , char K = 3 ; (2) r ∈ { , } , ℓ + m ... , char K = 3 . ( c ) dim K HH t ( R ) = 1 , if one of the following conditions takes place :(1) r ∈ { , } , t > , ℓ + m ... , char K = 3 ; (2) r ∈ { , } ; (3) r ∈ { , } , ℓ + m ... ; (4) r = 3 , ℓ + m ... or char K = 2 ; (5) r ∈ { , } , ℓ + m ... , char K = 2 ; (6) r ∈ { , } , ℓ + m ... , char K = 3 ; (7) r = 5 , char K = 3 ; (8) r = 7 , ℓ + m ... or char K = 2 . ( d ) In other cases dim K HH t ( R ) = 0 . Generators of HH ∗ ( R )For s > Y (1) t , Y (2) t , . . . Y (22) t , such that deg Y ( i ) t = t , 0 t <
11 deg σ and t satisfies conditions of (i)th item from the list on page 3. For s = 1 introduce theset of generators Y (1) t , Y (2) t , . . . Y (24) t , such that deg Y ( i ) t = t , 0 t <
11 deg σ and t satisfiesconditions of (i)th item from the list on page 3 for i
22 and t = 0 if i >
22. Now let us describethe matrixes of Y ( i ) t componentwisely. Denotation . Let us represent the degree t of the generator element in the form t = 11 ℓ + r (0 r κ : { w ∈ K [ Q s ] } → Z which returns a coefficient of σ ( w ); κ ℓ ( w ) returns a coefficient of σ ℓ ( w ).(1) Y (1) t is an (6 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = κ ℓ ( α j + m ) ) e j ⊗ e j , i = j ;0 , otherwise.If s j < s , then y ij = e j + s )+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If 3 s j < s , then y ij = e j + s )+2 ⊗ e j + s )+2 , i = j ;0 , otherwise. If 5 s j < s , then y ij = κ ℓ ( α j + m ) ) e j +3 ⊗ e j +3 , i = j ;0 , otherwise.(2) Y (2) t is an (6 s × s )-matrix with a single nonzero element: y , = κ ℓ ( α ) w → ⊗ e . (3) Y (3) t is an (7 s × s )-matrix with two nonzero elements: y , = w → ⊗ e and y ,s = w → s +1 ⊗ e . (4) Y (4) t is an (7 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = w j → j + s )+1 ⊗ e j , i = j ;0 , otherwise.If s j < s −
1, then y ij = 0 . If 5 s − j < s −
1, then y ij = − κ ℓ ( α j + m ) ) w j +2 → j +3 ⊗ e j +2 , i = j − s ;0 , otherwise.If 6 s − j < s , then y ij = 0 . (5) Y (5) t is an (6 s × s )-matrix with a single nonzero element: y , = κ ℓ ( α ) w → ⊗ e . (6) Y (6) t is an (8 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = w j → j +2 ⊗ e j , i = j ;0 , otherwise.If s j < s , then y ij = 0 . If 3 s j < s , then y ij = − κ ℓ ( α j + m ) ) w j +1 → j +3 ⊗ e j +1 , i = j − s ;0 , otherwise.If 4 s j < s , then y ij = − κ ℓ ( α j + m +1) ) w j +2 → j +1) ⊗ e j +2 , i = j − s, j < s − j = 6 s − , otherwise.If 6 s j < s , then y ij = w j +3 → j +1)+1 ⊗ e j +3 , i = 5 s + ( j ) s , j < s − j = 8 s − , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 27 (7) Y (7) t is an (8 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = w j → j + s )+2 ⊗ e j , i = ( j ) s ;0 , otherwise.If 2 s j < s , then y ij = w j +1 → j +3 ⊗ e j +1 , i = j − s ;0 , otherwise.If 4 s j < s , then y ij = w j +2 → j +1) ⊗ e j +2 , i = j − s ;0 , otherwise.If 6 s j < s , then y ij = 0 . (8) Y (8) t is an (9 s × s )-matrix with a single nonzero element: y , = w → ⊗ e . (9) Y (9) t is an (9 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = 0 . If s j < s , then y ij = κ ℓ ( α j + m ) ) e j ⊗ e j , i = j − s ;0 , otherwise.If 2 s j < s , then y ij = e j +1 ⊗ e j +1 , i = j − s ;0 , otherwise.If 4 s j < s , then y ij = 0 . If 5 s j < s , then y ij = e j + s )+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If 6 s j < s , then y ij = 0 . If 7 s j < s , then y ij = e j +2 ⊗ e j +2 , i = j − s ;0 , otherwise.If 8 s j < s , then y ij = − κ ℓ ( α j + m ) ) e j +3 ⊗ e j +3 , i = j − s ;0 , otherwise. (10) Y (10) t is an (9 s × s )-matrix with a single nonzero element: y ,s = κ ℓ ( α ) w → ⊗ e . (11) Y (11) t is an (8 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = w j → j +1 ⊗ e j , i = ( j ) s ;0 , otherwise.If 2 s j < s , then y ij = κ ℓ ( α j + m +1) ) w j +1 → j +1) ⊗ e j +1 , i = j − s ;0 , otherwise.If 4 s j < s , then y ij = − κ ℓ ( α j + m ) ) w j +2 → j +3 ⊗ e j +2 , i = j − s ;0 , otherwise.If 6 s j < s , then y ij = f ( j, s ) w j +3 → j +1)+2 ⊗ e j +3 , i = 5 s + ( j ) s ;0 , otherwise.(12) Y (12) t is an (8 s × s )-matrix with two nonzero elements: y s, s = κ ℓ ( α j + m ) ) w j +2 → j +3 ⊗ e j +2 and y s, s = κ ℓ ( α j + m ) ) w j +2 → j +3 ⊗ e j +2 . (13) Y (13) t is an (9 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = 0 . If s j < s , then y ij = e j + s )+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If 2 s j < s , then y ij = 0 . If 3 s j < s , then y ij = e j +1 ⊗ e j +1 , i = j − s ;0 , otherwise.If 4 s j < s , then y ij = 0 . If 5 s j < s , then y ij = e j + s )+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If 7 s j < s , then y ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 29 If 8 s j < s , then y ij = w j +3 → j +1) ⊗ e j +3 , i = j − s ;0 , otherwise.(14) Y (14) t is an (9 s × s )-matrix with a single nonzero element: y s, s = κ ℓ ( α j + m ) ) w j +3 → j +1)+3 ⊗ e j +3 . (15) Y (15) t is an (9 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = κ ℓ ( α j + m ) ) e j ⊗ e j , i = j ;0 , otherwise.If s j < s , then y ij = 0 . If 2 s j < s , then y ij = w j +1 → j +2 ⊗ e j +1 , i = j − s ;0 , otherwise.If 3 s j < s , then y ij = 0 . If 4 s j < s , then y ij = w j + s )+1 → j + s )+2 ⊗ e j + s )+1 , i = j − s ;0 , otherwise.If 5 s j < s , then y ij = 0 . If 7 s j < s , then y ij = κ ℓ ( α j + m ) ) e j +3 ⊗ e j +3 , i = j − s ;0 , otherwise.(16) Y (16) t is an (8 s × s )-matrix with two nonzero elements: y , = w → ⊗ e and y s, s = w → ⊗ e . (17) Y (17) t is an (8 s × s )-matrix with two nonzero elements: y , = w → s +2 ⊗ e and y ,s = w → ⊗ e . (18) Y (18) t is an (6 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = 0 . If s j < s , then y ij = w j + s )+1 → j + s )+2 ⊗ e j + s )+1 , i = j ;0 , otherwise.If 3 s j < s , then y ij = 0 . If 5 s j < s , then y ij = − κ ℓ ( α j + m +1) ) w j +3 → j +1) ⊗ e j +3 , i = j ;0 , otherwise.(19) Y (19) t is an (7 s × s )-matrix with two nonzero elements: y s,s = κ ℓ ( α j + m +1) ) w j + s )+1 → j +1) ⊗ e j + s )+1 and y s, s = κ ℓ ( α j + m +1) ) w j + s )+1 → j +1) ⊗ e j + s )+1 . (20) Y (20) t is an (7 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = κ ℓ ( α j + m ) ) w j → j +3 ⊗ e j , i = j ;0 , otherwise.If s j < s , then y ij = − κ ℓ ( α j + m +1) ) w j + s )+1 → j +1) ⊗ e j + s )+1 , i = j ;0 , otherwise.If 2 s j < s , then y ij = 0 . If 3 s j < s , then y ij = w j + s )+2 → j + s +1)+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If 4 s j < s , then y ij = 0 . If 6 s j < s , then y ij = w j +3 → j +1)+2 ⊗ e j +3 , i = j − s ;0 , otherwise.(21) Y (21) t is an (6 s × s )-matrix with a single nonzero element: y , = − κ ℓ ( α ) w → ⊗ e . (22) Y (22) t is an (6 s × s )-matrix, whose elements y ij have the following form:If 0 j < s , then y ij = κ ℓ ( α j + m ) ) e j ⊗ e j , i = j ;0 , otherwise.If s j < s , then y ij = 0 . If 5 s j < s , then y ij = − κ ℓ ( α j + m ) ) e j +3 ⊗ e j +3 , i = j ;0 , otherwise.(23) Y (23) t is an (6 s × s )-matrix with a single nonzero element: y , = w → ⊗ e . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 31 (24) Y (24) t is an (6 s × s )-matrix with a single nonzero element: y , = w → ⊗ e .
6. Ω -shifts of generators of the algebra HH ∗ ( R )Let Q • → R be the minimal projective bimodule resolution of the algebra R , constructed inparagraph 3. Any t -cocycle f ∈ Ker δ t is lifted (uniquely up to homotopy) to a chain map ofcomplexes { ϕ i : Q t + i → Q i } i > . The homomorphism ϕ i is called the i th translate of the cocycle f and will be denoted by Ω i ( f ). For cocycles f ∈ Ker δ t and f ∈ Ker δ t we have( ∗ ) cl f · cl f = cl(Ω ( f )Ω t ( f )) . We shall now describe Ω-translates for generators of the algebra HH ∗ ( R ) and then find multi-plications of the generators using the formula ( ∗ ). Denotations . (1) For generator degree t represent it in the form t = 11 ℓ + r (0 r t represent t in the form t = 11 ℓ + r (0 r Proposition 17 (Translates for the case 1) . (I) Let r ∈ N , r < . r -translates of the elements Y (1) t are described by the following way. (1) If r = 0 , then Ω ( Y (1) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + s )+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + s )+2 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j +3 ⊗ e j +3 , i = j ;0 , otherwise. (2) If r = 1 , then Ω ( Y (1) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m )+1 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+2 ⊗ e j +1 , i = j ;0 , otherwise. If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m )+3 ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j +3 , i = j ;0 , otherwise. (3) If r = 2 , then Ω ( Y (1) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j + m ) − ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j +3 , i = j ;0 , otherwise. (4) If r = 3 , then Ω ( Y (1) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m )+2 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m )+3 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m +1)+1 ⊗ e j +3 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 33 (5) If r = 4 , then Ω ( Y (1) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j + m ) − ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+1 ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+2 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) − ⊗ e j +3 , i = j ;0 , otherwise. (6) If r = 5 , then Ω ( Y (1) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m )+1 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j +1 , i = j ;0 , otherwise. If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m + s )+3 ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m +1)+2 ⊗ e j +3 , i = j ;0 , otherwise. (7) If r = 6 , then Ω ( Y (1) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+1 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+2 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) − ⊗ e j +3 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j +3 , i = j ;0 , otherwise. (8) If r = 7 , then Ω ( Y (1) t ) is described with (8 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 35 If j < s , then b ij = e j + m )+2 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m )+3 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m +1)+1 ⊗ e j +3 , i = j ;0 , otherwise. (9) If r = 8 , then Ω ( Y (1) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j + m ) − ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j +3 , i = j ;0 , otherwise. (10) If r = 9 , then Ω ( Y (1) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j + m )+3 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m +1) ⊗ e j + s )+1 , i = j ;0 , otherwise. If s j < s , then b ij = e j + m +1)+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m +1+ s )+2 ⊗ e j +3 , i = j ;0 , otherwise. (11) If r = 10 , then Ω ( Y (1) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+2 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m ) ) e j + m + s )+3 ⊗ e j +3 , i = j ;0 , otherwise. (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (1) t ) is a Ω r ( Y (1) t ) , whose left components twisted by σ ℓ . Proposition 18 (Translates for the case 2) . (I) Let r ∈ N , r < . Denote by κ = f ( s, κ ℓ ( α ) . Then r -translates of the elements Y (2) t are described by the following way. (1) If r = 0 , then Ω ( Y (2) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b , = κ ℓ ( α ) w j + m ) → j + m +1) ⊗ e j . (2) If r = 1 , then Ω ( Y (2) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b j, s +(5) s = κ w j + m +1) → j + m +2) ⊗ e j +3 . (3) If r = 2 , then Ω ( Y (2) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, s +(4) s = κ w j + m +1) → j + m +2) ⊗ e j +3 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 37 (4) If r = 3 , then Ω ( Y (2) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, s +(4) s = κ w j + m +1) → j + m +2) ⊗ e j +2 ; b j, s +(4) s = κ w j + m +1) → j + m +2) ⊗ e j +2 . (5) If r = 4 , then Ω ( Y (2) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b j,s +(4) s = κ w j + m ) → j + m +1) ⊗ e j . (6) If r = 5 , then Ω ( Y (2) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, s +(3) s = κ w j + m +1) → j + m +2) ⊗ e j +1 ; b j, s +(3) s = κ w j + m +1) → j + m +2) ⊗ e j +1 . (7) If r = 6 , then Ω ( Y (2) t ) is described with (9 s × s ) -matrix with the following two nonzeroelements : b j, (3) s = κ w j + m ) → j + m +1) ⊗ e j ; b j, s +(2) s = κ w j + m +1) → j + m +2) ⊗ e j +3 . (8) If r = 7 , then Ω ( Y (2) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, s +(2) s = κ w j + m +1) → j + m +2) ⊗ e j +2 ; b j, s +(2) s = κ w j + m +1) → j + m +2) ⊗ e j +2 . (9) If r = 8 , then Ω ( Y (2) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, s +(1) s = κ w j + m +1) → j + m +2) ⊗ e j +3 . (10) If r = 9 , then Ω ( Y (2) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b j,s +(1) s = κ w j + m +1) → j + m +2) ⊗ e j + s )+1 ; b j, s +(1) s = κ w j + m +1) → j + m +2) ⊗ e j + s )+1 . (11) If r = 10 , then Ω ( Y (2) t ) is described with (6 s × s ) -matrix with one nonzero elementthat is of the following form : b j, (1) s = κ w j + m ) → j + m +1) ⊗ e j . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (2) t ) is a Ω r ( Y (2) t ) , whose left components twisted by σ ℓ . Proposition 19 (Translates for the case 3) . (I) Let r ∈ N , r < . r -translates of the elements Y (3) t are described by the following way. (1) If r = 0 , then Ω ( Y (3) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b , = w j → j +1 ⊗ e j ; b ,s = w j → j +1 ⊗ e j . (2) If r = 1 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( s + j +1) s ,s +(1) s = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( s + j +1) s ,s +(1) s = e j + m + s +1)+2 ⊗ w j + s )+1 → j + s +1)+1 ; b ( s + j +1) s , s +(1) s = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( s + j +1) s , s +(1) s = e j + m + s +1)+2 ⊗ w j + s )+1 → j + s +1)+1 ; b ( j +1) s , s +(1) s = e j + m +1)+1 ⊗ w j + s )+2 → j +1) ; b ( j +1) s , s +(1) s = e j + m +1)+1 ⊗ w j + s )+2 → j +1) ; b j + s, s +(1) s = − κ ℓ ( α j + m ) ) w j + m +1) → j + m +2) ⊗ e j +3 . (3) If r = 2 , then Ω ( Y (3) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , s +(1) s = f (( j ) s , s − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +1 → j +1) ; b ( j +1) s , s +(1) s = − f (( j ) s , s − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +1 → j +1) ; b j − s, s +(1) s = κ ℓ ( α j + m ) ) w j + m + s )+1 → j + m +1) ⊗ e j +2 ; b j − s, s +(1) s = κ ℓ ( α j + m ) ) w j + m + s )+1 → j + m +1) ⊗ e j +2 . (4) If r = 3 , then Ω ( Y (3) t ) is described with (9 s × s ) -matrix with the following nonzeroelements : b j − s,s +(1) s = − κ ℓ ( α j + m ) ) w j + m + s )+2 → j + m +1) ⊗ e j ; b j + s,s +(1) s = − κ ℓ ( α j + m ) ) w j + m )+3 → j + m +1) ⊗ w j → j + s )+1 ; b j, s +(1) s = − w j + m )+3 → j + m + s +1)+1 ⊗ e j +1 ; b j, s +(1) s = − w j + m )+3 → j + m + s +1)+1 ⊗ e j +1 . (5) If r = 4 , then Ω ( Y (3) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b j, (1) s = w j + m − → j + m )+1 ⊗ e j ; b s +( j +1) s , s = f (( j ) s , s − κ ℓ ( α j + m ) ) e j + m +1) ⊗ w j +1 → j +1) ; b s +( j +1) s , s = − f (( j ) s , s − κ ℓ ( α j + m ) ) e j + m +1) ⊗ w j +1 → j +1) ; b s +( s + j +1) s , s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j + s +1)+1 ; b s +( s + j +1) s − f ( s, s, s = e j + m +1)+2 ⊗ w j +3 → j + s +1)+2 ; b s +( s + j +1) s , s = w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j + s +1)+1 ; b s +( s + j +1) s + f ( s, s, s = − e j + m +1)+2 ⊗ w j +3 → j + s +1)+2 . (6) If r = 5 , then Ω ( Y (3) t ) is described with (9 s × s ) -matrix with the following nonzeroelements : b ( s + j +1) s ,s = − e j + m + s +1)+1 ⊗ w j + s )+1 → j +1) ; b ( s + j +1) s , s = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +1 → j +1) ; b ( j +1) s , s = − e j + m +1)+1 ⊗ w j +1 → j +1) ; b ( j +1) s , s = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b ( s + j +1) s , s = κ ℓ ( α j + m ) ) w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) ; b ( j +1) s , s = κ ℓ ( α j + m ) ) w j + m +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) ; b s +( s + j +1) s , s = − κ ℓ ( α j + m ) ) e j + m +1)+3 ⊗ w j +3 → j + s +1)+2 ; b s +( j +1) s , s = − κ ℓ ( α j + m ) ) e j + m +1)+3 ⊗ w j +3 → j +1)+2 ; b s +( s + j +1) s , s = − κ ℓ ( α j + m ) ) e j + m +2) ⊗ w j +3 → j + s +1)+1 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 39 (7) If r = 6 , then Ω ( Y (3) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , s = − κ ℓ ( α j + m ) ) e j + m +1) ⊗ w j +2 → j +1) ; b ( j +1) s , s = w j + m +1) → j + m +1)+1 ⊗ w j +3 → j +1) ; b s +( j +1) s + f ( s, s, s = − e j + m +1)+1 ⊗ w j +3 → j +1)+1 ; b s +( j +1) s − f ( s, s, s = − e j + m +1)+1 ⊗ w j +3 → j +1)+1 . (8) If r = 7 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j + s +1) s ,s = − e j + m + s +1)+2 ⊗ w j + s )+1 → j +1) ; b ( j + s +1) s , s = − e j + m + s +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s = κ ℓ ( α j + m ) ) w j + m +1+ f ( s, s )+2 → j + m +2) ⊗ w j +3 → j +1) ; b s +( j +1) s , s = − κ ℓ ( α j + m ) ) w j + m +1)+3 → j + m +2) ⊗ w j +3 → j + s +1+ f ( s, s )+1 ; b s +( j + s +1) s , s = − κ ℓ ( α j + m ) ) e j + m +2) ⊗ w j +3 → j + s +1)+2 ; b s +( j +1) s , s = − κ ℓ ( α j + m ) ) e j + m +2) ⊗ w j +3 → j +1)+2 . (9) If r = 8 , then Ω ( Y (3) t ) is described with (7 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , = f (( j ) s , s − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j → j +1) ; b ( j +1) s ,s + f ( s, s = − κ ℓ ( α j + m ) ) w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +1) ; b s +( j + s +1) s , s = e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 ; b ( j +1) s , s + f ( s, s = w j + m )+3 → j + m +1)+1 ⊗ w j + s )+2 → j +1) ; b s +( j + s +1) s , s = e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 ; b j, s = w j + m +1) → j + m + s +1)+2 ⊗ e j +3 ; b j − s, s = w j + m +1) → j + m + s +1)+2 ⊗ e j +3 . (10) If r = 9 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following two nonzeroelements : b j, s = − w j + m +1)+1 → j + m +1)+2 ⊗ e j + s )+2 ; b j, s = − w j + m +1)+1 → j + m +1)+2 ⊗ e j + s )+2 . (11) If r = 10 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , = f (( j ) s , s − κ ℓ ( α j + m ) ) e j + m +1) ⊗ w j → j +1) ; b j +3 s, = − κ ℓ ( α j + m ) ) w j + m + s )+2 → j + m +1) ⊗ w j → j +2 ; b j +4 s, = κ ℓ ( α j + m ) ) w j + m )+2 → j + m +1) ⊗ w j → j + s )+2 ; b j +5 s, = − κ ℓ ( α j + m ) ) w j + m )+3 → j + m +1) ⊗ w j → j +3 ; b j,s = w j + m )+1 → j + m +1)+1 ⊗ e j + s )+1 ; b j, s = w j + m )+1 → j + m +1)+1 ⊗ e j + s )+1 ; b s +( j + s +1) s , s +( s − s = e j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 ; b s +( j + s +1) s , s +( s − s = e j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 ; b s +( j +1) s , s +( s − s = − κ ℓ ( α j + m ) ) w j + m + s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1)+2 ; b s +( j + s +1) s , s +( s − s = κ ℓ ( α j + m ) ) w j + m +1)+2 → j + m +1)+3 ⊗ w j +3 → j + s +1)+2 ; b s +( j +1) s , s +( s − s = f (( j ) s , s − κ ℓ ( α j + m ) ) e j + m +1)+3 ⊗ w j +3 → j +1)+3 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (3) t ) is a Ω r ( Y (3) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 20 (Translates for the case 4) . (I) Let r ∈ N , r < . r -translates of the elements Y (4) t are described by the following way. (1) If r = 0 , then Ω ( Y (4) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) → j + m + s )+1 ⊗ e j , i = ( j ) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s − , then b ij = − κ ℓ ( α j + m ) ) w j + m )+2 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s − j < s , then b ij = 0 . (2) If r = 1 , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m )+1 → j + m )+3 ⊗ e j , i = j ; κ w j + m )+2 → j + m )+3 ⊗ w j → j +1 , i = j + 2 s ; κ e j + m )+3 ⊗ w j → j +2 , i = j + 4 s, j < s − − κ e j + m )+3 ⊗ w j → j + s )+2 , i = j + 5 s, j < s − , otherwise,where κ = κ ℓ ( α j + m ) ) . If s j < s − , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ; − w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 3 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 41 If s j < s − , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ; − w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = f ( j, s − κ w j + m +1) → j + m +2) ⊗ e j +3 , i = j + s ; f ( s, κ w j + m +1)+3 → j + m +2) ⊗ w j +3 → j + s +1+ sf ( s, ,i = 4 s + ( j + 1) s , j = 5 s + ( s − s ; f ( s, κ e j + m +2) ⊗ w j +3 → j + s +1)+3 , i = 6 s + ( j + 1) s , j = 5 s + ( s − s ;0 , otherwise,where κ = κ ℓ ( α j + m +2) ) . (3) If r = 2 , then Ω ( Y (4) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s − , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = − w j + m + s ) − → j + m + s )+2 ⊗ e j , i = j − s ;0 , otherwise.If s − j < s , then b ij = − e j + m + s )+2 ⊗ w j → j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s ; − κ ℓ ( α j + m ) ) w j + m )+2 → j + m )+3 ⊗ e j +1 , i = j − s, j = 3 s − , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s ; − κ ℓ ( α j + m ) ) w j + m )+2 → j + m )+3 ⊗ e j +1 , i = j − s, j < s − , otherwise. If s j < s − , then b ij = − κ w j + m )+3 → j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s ; − κ w j + m + s )+1 → j + m +1) ⊗ e j +2 , i = j − s ;0 , otherwise,where κ = κ ℓ ( α j + m +1) ) .If s − j < s − , then b ij = 0 .If s − j < s , then b ij = κ w j + m )+3 → j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s ; κ w j + m + s )+1 → j + m +1) ⊗ e j +2 , i = j − s ;0 , otherwise,where κ = − κ ℓ ( α j + m +1) ) .If s j < s − , then b ij = e j + m + s +1)+1 ⊗ w j +3 → j +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m + s +1)+1 ⊗ w j +3 → j +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise. (4) If r = 3 , then Ω ( Y (4) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s − , then b ij = κ ℓ ( α j + m ) ) w j + m )+2 → j + m )+3 ⊗ e j , i = j ; κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j → j +1 , i = j + 2 s ; κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j → j + s )+1 , i = j + 3 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = κ ℓ ( α j + m +1) ) w j + m + s )+2 → j + m +1) ⊗ e j , i = j − s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m )+3 → j + m +1)+1 ⊗ e j +1 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 43 If s j < s − , then b ij = − w j + m )+3 → j + m +1)+1 ⊗ e j +1 , i = j ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ e j +2 , i = j ; − e j + m + s +1)+1 ⊗ w j +2 → j +3 , i = j + 3 s ;0 , otherwise.If j < , then b ij = − e j + m +1)+2 ⊗ w j +2 → j +1) , i = ( j + 1) s , s = 1;0 , otherwise.If s − j < s − , then b ij = − e j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , s > , otherwise.If s − j < s , then b ij = − w j + m +1) → j + m +1)+1 ⊗ e j + s )+2 , i = j − s ; − e j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = j ;0 , otherwise.If s j < s + ( s − s , then b ij = 0 .If s + ( s − s j < s + ( s − s + 1 , then b ij = κ ℓ ( α j + m ) ) e j + m +1)+3 ⊗ w j +3 → j +1+ sf ( s, , i = 2 s + ( j + 1) s ;0 , otherwise.If s + ( s − s + 1 j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (4) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s − , then b ij = − e j + m + s )+1 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m + s ) − → j + m + s )+1 ⊗ e j , i = j − s ; − e j + m + s )+1 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise. If s j < s − (2) s , then b ij = f ( s, κ ℓ ( α j + m +1) ) e j + m +1) ⊗ w j +1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s − (2) s j < s − (2) s , then b ij = 0 .If s − (2) s j < s , then b ij = f ( j, s − κ ℓ ( α j + m +1) ) e j + m +1) ⊗ w j +1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +2 → j +1) , i = ( j + 1) s ; − κ ℓ ( α j + m ) ) w j + m + s )+2 → j + m )+3 ⊗ e j +2 , i = j + s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +2 → j +1) , i = ( j + 1) s ; − κ ℓ ( α j + m ) ) w j + m + s )+2 → j + m )+3 ⊗ e j +2 , i = j + 2 s ;0 , otherwise.If s j < s − , then b ij = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1)+1 , i = 2 s + ( j + 1) s ; e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 5 s + ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1)+1 , i = 2 s + ( j + 1) s ; − e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 5 s + ( j + 1) s ;0 , otherwise. (6) If r = 5 and s = 1 , then Ω ( Y (4) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b j +3 s, = − κ ℓ ( α j + m +1) ) e j + m ) ⊗ w j → j +1)+1 ; b j − s,s = − e j + m +1)+1 ⊗ w j +1)+1 → j ; b j − s, s = w j + m +1)+1 → j + m +1)+2 ⊗ w j +1 → j ; b j − s, s = w j + m )+3 → j + m +1)+2 ⊗ e j +1)+2 ; b j, s = − e j + m +1)+2 ⊗ w j +1)+2 → j +3 ; b j − s, s = κ w j + m +1)+1 → j + m )+3 ⊗ w j +3 → j ; b j − s, s = κ w j + m )+1 → j + m )+3 ⊗ w j +3 → j ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 45 b j − s, s = − κ e j + m )+3 ⊗ w j +3 → j +1)+2 ; b j − s, s = − κ e j + m )+3 ⊗ w j +3 → j +2 , where κ = − κ ℓ ( α s + m +1) ) . (7) If r = 5 and s > , then Ω ( Y (4) t ) is described with (9 s × s ) -matrix with the followingelements b ij : If j < s , then b ij = κ e j + m +1) ⊗ w j → j +1 , i = j + 2 s ; κ w j + m )+1 → j + m +1) ⊗ e j , i = j, j = s − κ w j + m + s )+1 → j + m +1) ⊗ e j , i = j + s, j = s − − κ w j + m )+3 → j + m +1) ⊗ w j → j +2 , i = j + 4 s, j = s − − κ w j + m )+3 → j + m +1) ⊗ w j → j + s )+2 , i = j + 5 s, j = s − , otherwise,where κ = κ ℓ ( α j + m +1) ) .If s j < s − , then b ij = − e j + m +1)+1 ⊗ w j + s )+1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s − , then b ij = − e j + m + s +1)+1 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = − e j + m + s +1)+1 ⊗ w j +1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 . If s − j < s − , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = w j + m )+3 → j + m +1)+2 ⊗ e j + s )+2 , i = j − s ; e j + m +1)+2 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m )+3 → j + m +1)+2 ⊗ e j + s )+2 , i = j − s ; − e j + m +1)+2 ⊗ w j + s )+2 → j +3 , i = j ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s − , then b ij = κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; κ w j + m +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; − κ e j + m +1)+3 ⊗ w j +3 → j + s +1)+2 , i = 4 s + ( j + 1) s ; − κ e j + m +1)+3 ⊗ w j +3 → j +1)+2 , i = 5 s + ( j + 1) s ;0 , otherwise,where κ = κ ℓ ( α j + m +1) ) .If s − j < s , then b ij = 0 .If s j < s − , then b ij = − κ ℓ ( α j + m +2) ) e j + m +2) ⊗ w j +3 → j +1)+1 , i = 2 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 . (8) If r = 6 and s = 1 , then Ω ( Y (4) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j +3 s, = w j + m +1)+1 → j + m +1)+2 ⊗ w j → j +1)+1 ; b j +3 s,s = − e j + m +1)+2 ⊗ w j → j +1 ; b j, s = κ ℓ ( α j + m ) ) w j + m )+1 → j + m )+3 ⊗ e j +1 ; b j + s, s = − κ ℓ ( α j + m ) ) w j + m +1)+2 → j + m )+3 ⊗ e j +1 ; b j + s, s = − κ ℓ ( α j + m ) ) w j + m )+2 → j + m ) ⊗ e j +2 ; b j +2 s, s = κ ℓ ( α j + m ) ) w j + m )+3 → j + m ) ⊗ w j +2 → j +3 ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 47 b j − s, s = − w j + m ) → j + m )+1 ⊗ w j +3 → j ; b j − s, s = e j + m )+1 ⊗ w j +3 → j +1 . (9) If r = 6 and s > , then Ω ( Y (4) t ) is described with (8 s × s ) -matrix with the followingelements b ij : If j < s , then b ij = − w j + m ) → j + m + s )+2 ⊗ e j , i = j ; w j + m + s )+1 → j + m + s )+2 ⊗ w j → j + s )+1 , i = j + 3 s ; − e j + m + s )+2 ⊗ w j → j + s )+2 , i = j + 6 s, j < s − , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ w j → j + s )+1 , i = j ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − e j + m + s )+2 ⊗ w j → j + s )+2 , i = j + 4 s ;0 , otherwise.If s j < s − , then b ij = κ ℓ ( α j + m ) ) w j + m )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + 3 s ; − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +1 → j +3 , i = j + 5 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = κ ℓ ( α j + m ) ) w j + m )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + 3 s ; − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j +1 → j +3 , i = j + 4 s ;0 , otherwise.If s j < s − , then b ij = κ e j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s ; κ w j + m )+2 → j + m +1) ⊗ e j +2 , i = j + s ; − κ w j + m )+3 → j + m +1) ⊗ w j +2 → j +3 , i = j + 3 s ;0 , otherwise,where κ = κ ℓ ( α j + m +1) ) .If s − j < s − , then b ij = 0 . If s − j < s , then b ij = − κ ℓ ( α j + m ) ) e j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s ; − κ ℓ ( α j + m ) ) w j + m )+2 → j + m +1) ⊗ e j +2 , i = j + s ; κ ℓ ( α j + m ) ) w j + m )+3 → j + m +1) ⊗ w j +2 → j +3 , i = j + 2 s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m + s +1)+1 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = e j + m + s +1)+1 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 . (10) If r = 7 and s = 1 , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j − s,s = − e j + m +1)+2 ⊗ w j +1)+1 → j ; b j + s, s = w j + m )+3 → j + m +1)+2 ⊗ e j +1)+1 ; b j + s, s = − w j + m ) → j + m )+1 ⊗ e j +1)+2 ; b j +3 s, s = e j + m )+1 ⊗ w j +1)+2 → j +3 ; b j − s, s = κ w j + m +1)+2 → j + m ) ⊗ w j +3 → j ; b j − s, s = − κ e j + m ) ⊗ w j +3 → j +1)+2 ; b j, s = − κ e j + m ) ⊗ w j +3 → j +2 , where κ = − κ ℓ ( α s + m ) ) . (11) If r = 7 and s > , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the followingelements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) w j + m + s )+2 → j + m )+3 ⊗ e j , i = j + s ; − κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ; e j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 3 s ;0 , otherwise.If s − j < s − , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 49 If s − j < s , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ; e j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s − , then b ij = κ e j + m +2) ⊗ w j +3 → j + s +1)+2 , i = 4 s + ( j + 1) s ; κ e j + m +2) ⊗ w j +3 → j +1)+2 , i = 5 s + ( j + 1) s ;0 , otherwise,where κ = κ ℓ ( α j + m +2) ) .If s − j < s , then b ij = 0 . (12) If r = 8 and s = 1 , then Ω ( Y (4) t ) is described with (7 s × s ) -matrix with the followingnonzero elements : b j +2 s, = κ ℓ ( α j + m ) ) w j + m +1)+2 → j + m )+3 ⊗ w j → j +1)+1 ; b j, s = κ ℓ ( α j + m ) ) w j + m +1)+2 → j + m ) ⊗ e j +1)+1 ; b j − s, s = − w j + m )+3 → j + m )+1 ⊗ w j +1)+2 → j ; b j + s, s = − w j + m ) → j + m )+1 ⊗ w j +1)+2 → j +3 ; b j − s, s = − w j + m )+3 → j + m +1)+2 ⊗ w j +3 → j ; b j − s, s = e j + m +1)+2 ⊗ w j +3 → j +1)+1 ; b j − s, s = w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j +2 ; b j − s, s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j +2 . (13) If r = 8 and s > , then Ω ( Y (4) t ) is described with (7 s × s ) -matrix with the followingelements b ij : If j < s , then b ij = κ ℓ ( α j + m ) ) e j + m )+3 ⊗ w j → j +1) , i = ( j + 1) s ; κ ℓ ( α j + m ) ) w j + m )+2 → j + m )+3 ⊗ w j → j +1 , i = j + s, j < s − κ ℓ ( α j + m ) ) w j + m − → j + m )+3 ⊗ e j , i = j, j = s − κ ℓ ( α j + m ) ) w j + m + s )+2 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 2 s, j = s − , otherwise.If s j < s − , then b ij = κ w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; κ w j + m + s )+2 → j + m +1) ⊗ e j + s )+1 , i = j ;0 , otherwise,where κ = − κ ℓ ( α j + m +1) ) .If s − j < s − , then b ij = 0 . If s − j < s , then b ij = − κ w j + m )+1 → j + m +1) ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ; κ e j + m +1) ⊗ w j + s )+1 → j +3 , i = j + 3 s ;0 , otherwise,where κ = κ ℓ ( α j + m +1) ) .If s j < s − , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m +1)+2 ⊗ w j +3 → j +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s − , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j + s +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = e j + m +1)+2 ⊗ w j +3 → j +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s − , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j + s +1)+2 , i = 4 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 . (14) If r = 9 and s = 1 , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j, s = − w j + m ) → j + m )+1 ⊗ e j +1)+1 ; b j +2 s, s = e j + m )+2 ⊗ w j +1)+2 → j +3 ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 51 b j, s = − κ ℓ ( α j + m +1) ) w j + m )+2 → j + m )+3 ⊗ e j +3 ; b j + s, s = κ ℓ ( α j + m +1) ) w j + m +1)+2 → j + m )+3 ⊗ e j +3 . (15) If r = 9 and s > , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the followingelements b ij : If s − j < s , then b ij = − κ w j + m )+3 → j + m +1) ⊗ e j , i = j ; − κ e j + m +1) ⊗ w j → j +1 , i = j + s ;0 , otherwise,where κ = κ ℓ ( α j + m +1) ) .If s j < s − , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = e j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s − , then b ij = κ ℓ ( α j + m ) ) e j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 . (16) If r = 10 and s = 1 , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j +2 s, = κ w j + m )+1 → j + m ) ⊗ w j → j +1)+1 ; b j +4 s, = − κ w j + m )+2 → j + m ) ⊗ w j → j +1)+2 ; b j − s,s = − w j + m ) → j + m )+1 ⊗ w j +1)+1 → j ; b j +4 s,s = − w j + m )+3 → j + m )+1 ⊗ w j +1)+1 → j +3 ; b j − s, s = − w j + m ) → j + m )+2 ⊗ w j +1)+2 → j ; b j +2 s, s = − w j + m )+3 → j + m )+2 ⊗ w j +1)+2 → j +3 ; b j − s, s = − κ w j + m +1)+1 → j + m )+3 ⊗ w j +3 → j +1 ; b j − s, s = κ w j + m +1)+2 → j + m )+3 ⊗ w j +3 → j +2 , where κ = κ ℓ ( α m +1) ) , κ = κ ℓ ( α s + m +1) ) . (17) If r = 10 and s > , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the followingelements b ij : If j < s , then b ij = κ w j + m ) → j + m +1) ⊗ e j , i = j ; κ w j + m )+1 → j + m +1) ⊗ w j → j + s )+1 , i = j + 2 s ; − κ w j + m )+2 → j + m +1) ⊗ w j → j + s )+2 , i = j + 4 s ; κ e j + m +1) ⊗ w j → j +1) , i = ( j + 1) s , j = s − κ w j + m )+3 → j + m +1) ⊗ w j → j +3 , i = j + 5 s, j = s − , otherwise,where κ = κ ℓ ( α j + m +1) ) .If s j < s − , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; − w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +3 , i = j + 4 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s − , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; − w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 53 If s j < s , then b ij = − κ w j + m +1) → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; − κ w j + m + s +1+ j )+1 → j + m +1)+3 ⊗ w j +3 → j +1+ j )+1 , i = 2 s + ( j + 1) s ; κ w j + m + s +1+ j )+2 → j + m +1)+3 ⊗ w j +3 → j +1+ j )+2 , i = 4 s + ( j + 1) s ; − f ( j, s − κ w j + m )+3 → j + m +1)+3 ⊗ e j +3 , i = j ; − κ e j + m +1)+3 ⊗ w j +3 → j +1)+3 , i = 5 s + ( j + 1) s , j = 6 s − , otherwise,where κ = κ ℓ ( α j + m +1) ) , j = sf ( j, s − Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (4) t ) is a Ω r ( Y (4) t ) , whose left components twisted by σ ℓ . Proposition 21 (Translates for the case 5) . (I) Let r ∈ N , r < . Denote by κ = κ ℓ ( α s ) ) .Then r -translates of the elements Y (5) t are described by the following way. (1) If r = 0 , then Ω ( Y (5) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b , = κ ℓ ( α ) w j → j +3 ⊗ e j . (2) If r = 1 , then Ω ( Y (5) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b ( j + s +1) s , s +(2) s = f (( j ) s , s − κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) ; b ( j + s +1) s , s +(2) s = − f (( j ) s , s − κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) ; b s +( j ) s , s +(2) s + sf ( s, sf ( s, = − κ ℓ ( γ j +1+ m ) κ w j + m +1) → j + m +2) ⊗ w j +2 → j +3 . (3) If r = 2 , then Ω ( Y (5) t ) is described with (9 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s , (2) s = κ e j + m )+3 ⊗ w j → j +1) ; b j + s,s +(2) s = f (( j ) s , s − κ ℓ ( γ j + m ) κ w j + m )+2 → j + m +1) ⊗ w j → j +1 . (4) If r = 3 , then Ω ( Y (5) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j +3 s, (2) s = − κ ℓ ( α j +1+ m ) ) κ ℓ ( γ j + m ) f (( j ) s , s − κ w j + m )+3 → j + m + s +1)+1 ⊗ w j → j + s )+1 ; b s +( j +1) s , s − s = f (( j ) s , ( s − s ) κ ℓ ( γ j +1+ m ) κ w j + m +1)+2 → j + m +2) ⊗ w j +1 → j +1) . (5) If r = 4 , then Ω ( Y (5) t ) is described with (9 s × s ) -matrix with the following nonzeroelements : b j +3 s, (1) s = κ ℓ ( γ j + m ) κ w j + m )+1 → j + m ) ⊗ w j → j +1)+1 , s = 1; b j +7 s, (1) s = − κ ℓ ( γ j + m ) κ w j + m )+2 → j + m ) ⊗ w j → j +1)+2 , s = 1; b ( j +1) s , (1) s = − f (( j ) s , s − κ ℓ ( γ j + m ) κ w j + m )+3 → j + m +1) ⊗ w j → j +1) , s > b s +( j +1) s , (1) s = f (( j ) s , s − κ ℓ ( γ j + m ) κ e j + m +1) ⊗ w j → j +1) , s > b ( j +1) s , s +1 − f ( s, = f ( s, f (( j ) s , s − κ ℓ ( γ j + m ) κ ℓ ( α j +1+ m ) ) κ w j + m )+3 → j + m + s +1+ sf ( s, ⊗ w j + s + sf ( s, → j +1) ; b s +( j +1) s , s +(1) s = − f (( j ) s , ( s − s ) κ ℓ ( γ j + m ) κ w j + m + s +1+ sf ( s, → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( s, ; b s +( j +1) s , s +(1) s = f (( j ) s , ( s − s ) κ ℓ ( γ j + m ) κ w j + m + s +1+ sf ( s, → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( s, . (6) If r = 5 and s = 1 , then Ω ( Y (5) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j + s,s = κ ℓ ( α j + m ) ) κ w j + m ) → j + m )+2 ⊗ w j → j +1)+1 ; b j +2 s,s = κ ℓ ( α j + m ) ) κ w j + m ) → j + m )+2 ⊗ w j → j +1 ; b j − s, s = κ w j + m )+1 → j + m )+3 ⊗ w j +1 → j ; b j − s, s = κ w j + m +1)+1 → j + m )+3 ⊗ w j +1 → j ; b j − s, s = − κ ℓ ( α j + m ) ) κ w j + m ) → j + m +1)+1 ⊗ w j +3 → j +1)+1 . (7) If r = 5 and s > , then Ω ( Y (5) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b s +( j +1) s , = − κ ℓ ( α j + m ) ) f (( j ) s , s − κ w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j → j +1) ; b ( j +1) s ,s +1 = κ ℓ ( α j + m ) ) f (( j ) s , s − κ w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j → j +1) ; b ( j +1) s , s +1 = κ κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) ; b s +( j +1) s , s +1 = κ κ w j + m +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) ; b s +( j +1) s , s +1 = κ κ e j + m +2) ⊗ w j +2 → j +1)+1 , where κ = f (1 , s − κ ℓ ( γ s +1+ m ) , κ = κ ℓ ( γ s +1+ m ) κ ℓ ( γ s +2+ m ) f (1 , s − . (8) If r = 6 and s = 1 , then Ω ( Y (5) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j +2 s, = − κ w j + m +1)+2 → j + m )+3 ⊗ w j → j +1 ; b j +3 s, = − κ w j + m +1)+1 → j + m )+3 ⊗ w j → j +1)+1 ; b j +5 s, = − κ w j + m )+2 → j + m )+3 ⊗ w j → j +2 ; b j +7 s, = κ e j + m )+3 ⊗ w j → j +3 ; b j − s,s = − κ ℓ ( α j + m ) ) κ w j + m ) → j + m +1)+2 ⊗ w j +1)+1 → j ; b j − s, s = κ w j + m +1)+1 → j + m ) ⊗ w j +3 → j +1)+1 ; b j − s, s = − κ w j + m )+1 → j + m ) ⊗ w j +3 → j +1 . (9) If r = 6 and s > , then Ω ( Y (5) t ) is described with (6 s × s ) -matrix with the followingtwo nonzero elements : b ( j +1) s , s +1 = f (( j ) s , s − κ ℓ ( α j +2+ m ) ) κ w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s +1 = − f (( j ) s , s − κ ℓ ( α j +2+ m ) ) κ e j + m + s +1)+1 ⊗ w j + s )+2 → j + s +1)+1 . (10) If r = 7 and s = 1 , then Ω ( Y (5) t ) is described with (7 s × s ) -matrix with the followingnonzero elements : b j +4 s, = − κ w j + m ) → j + m )+3 ⊗ w j → j +2 ; b j +5 s, = − κ w j + m ) → j + m )+3 ⊗ w j → j +1)+2 ; b j,s = κ w j + m )+2 → j + m ) ⊗ w j +1)+1 → j ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 55 b j − s, s = − κ w j + m )+2 → j + m ) ⊗ w j +1)+1 → j ; b j − s, s = κ w j + m +1)+2 → j + m ) ⊗ w j +1)+1 → j ; b j − s, s = − κ ℓ ( α j +2+ m ) ) κ w j + m )+2 → j + m +1)+1 ⊗ w j +1)+2 → j ; b j +2 s, s = κ ℓ ( α j +2+ m ) ) κ w j + m ) → j + m +1)+1 ⊗ w j +1)+2 → j +2 ; b j − s, s = κ ℓ ( α j +2+ m ) ) κ w j + m )+2 → j + m +1)+1 ⊗ w j +1)+2 → j ; b j − s, s = κ ℓ ( α j +2+ m ) ) κ w j + m )+3 → j + m )+2 ⊗ w j +3 → j +1)+1 . (11) If r = 7 and s > , then Ω ( Y (5) t ) is described with (7 s × s ) -matrix with the followingnonzero elements : b ( j +1) s , = − κ ℓ ( γ j +1+ m ) f (( j ) s , s − κ w j + m +1)+2 → j + m +1)+3 ⊗ w j → j +1) ; b s +( j +1) s , s +1 = f (( j ) s , s − κ w j + m + s +1)+2 → j + m +2) ⊗ w j + s )+1 → j +1) ; b s +( j ) s , s +1 = − f (( j ) s , s − κ ℓ ( α j +2+ m ) ) κ w j + m + s +1)+1 → j + m + s +2)+1 ⊗ w j + s )+2 → j +3 . (12) If r = 8 and s = 1 , then Ω ( Y (5) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j +2 s, = − κ w j + m +1)+2 → j + m ) ⊗ w j → j +1)+1 ; b j +3 s, = − κ w j + m +1)+1 → j + m ) ⊗ w j → j +2 ; b j − s, s = − κ ℓ ( α j +1+ m ) ) κ w j + m )+3 → j + m )+2 ⊗ w j +1)+2 → j ; b j − s, s = − κ ℓ ( α j +1+ m ) ) κ e j + m )+2 ⊗ w j +1)+2 → j +1 ; b j − s, s = − κ ℓ ( α j +1+ m ) ) κ w j + m )+3 → j + m )+2 ⊗ w j +1)+2 → j ; b j − s, s = κ w j + m +1)+2 → j + m )+3 ⊗ w j +3 → j +1)+1 ; b j − s, s = κ w j + m +1)+1 → j + m )+3 ⊗ w j +3 → j +2 . (13) If r = 8 and s > , then Ω ( Y (5) t ) is described with (6 s × s ) -matrix with the followingtwo nonzero elements : b ( j +1) s , = − f (1 , s − κ w j + m )+3 → j + m +1) ⊗ w j → j +1) ; b j, s +1 = f (( j ) s , s − κ ℓ ( α j +1+ m ) ) κ w j + m + s )+2 → j + m + s +1)+1 ⊗ e j + s )+1 . (14) If r = 9 and s = 1 , then Ω ( Y (5) t ) is described with (6 s × s ) -matrix with the followingtwo nonzero elements : b j − s, s = κ ℓ ( α j + m ) ) κ w j + m ) → j + m +1)+1 ⊗ w j +1)+1 → j +1 ; b j +2 s, s = κ ℓ ( α j + m ) ) κ w j + m )+2 → j + m +1)+2 ⊗ w j +1)+2 → j +3 . (15) If r = 9 and s > , then Ω ( Y (5) t ) is described with (6 s × s ) -matrix with the followingtwo nonzero elements : b ( j +1) s , s = κ κ w j + m +1)+3 → j + m + s +2)+1 ⊗ w j + s )+1 → j +1) ; b ( j +1) s , s = κ κ w j + m +1)+3 → j + m + s +2)+2 ⊗ w j + s )+2 → j +1) , where κ = f ( s, κ ℓ ( α m +2) ) . (16) If r = 10 and s = 1 , then Ω ( Y (5) t ) is described with (7 s × s ) -matrix with the followingtwo nonzero elements : b j − s, s = − κ ℓ ( α j +1+ m ) ) κ w j + m ) → j + m +1)+2 ⊗ w j +1 → j ; b j +2 s, s = − κ ℓ ( α j +1+ m ) ) κ w j + m )+3 → j + m +1)+2 ⊗ w j +1 → j +3 . (17) If r = 10 and s > , then Ω ( Y (5) t ) is described with (7 s × s ) -matrix with the followingtwo nonzero elements : b ( j +1) s ,s = − f (( j ) s , s − f (( j ) s , s − κ ℓ ( α j +1+ m ) ) κ w j + m +1) → j + m +1)+1 ⊗ w j → j +1) ; b s +( j ) s , s = f (( j ) s , s − κ w j + m )+3 → j + m +1)+3 ⊗ w j +2 → j +3 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (5) t ) is a Ω r ( Y (5) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 22 (Translates for the case 6) . (I) Let r ∈ N , r < . r -translates of the elements Y (6) t are described by the following way. (1) If r = 0 , then Ω ( Y (6) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j → j +2 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m +1) ) w j +1 → j +3 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) w j +2 → j +1) ⊗ e j +2 , i = j − s, j < s − or j = 6 s − , otherwise.If s j < s , then b ij = w j +3 → j +1)+1 ⊗ e j +3 , i = 5 s + ( j ) s , j < s − or j = 8 s − , otherwise. (2) If r = 1 , then Ω ( Y (6) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ ( α j + m +1) ) w j + m + s )+1 → j + m )+3 ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) w j + m + s )+1 → j + m +1) ⊗ e j , i = j − s ; − κ ℓ ( α j + m +2) ) w j + m + s )+2 → j + m +1) ⊗ w j → j + s )+1 , i = j + s ;0 , otherwise.If s j < s − , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 57 If s − j < s , then b ij = − κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1) ⊗ w j → j + s )+2 , i = j + 3 s ; κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1) ⊗ w j → j +2 , i = j + 4 s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ w j +1 → j +2 , i = j + 2 s ;0 , otherwise.If s j < s − , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ w j +1 → j +2 , i = j + 2 s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = − w j + m )+3 → j + m +1)+1 ⊗ e j +2 , i = j ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m +1)+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If s j < s − , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +2 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m + s +1)+2 ⊗ e j +2 , i = j − s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m +1) → j + m +1)+3 ⊗ e j +3 , i = j − s ;0 , otherwise. (3) If r = 2 , then Ω ( Y (6) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − w j + m ) − → j + m )+1 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1) ⊗ w j +1 → j +1) ,i = ( j + 1) s , j < s − or j = 4 s − , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +1) ) w j + m + s )+1 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +1) ) w j + m + s )+1 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (4) If r = 3 , then Ω ( Y (6) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m )+2 → j + m +1) ⊗ e j , i = j ; κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m + s +1)+2 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = − e j + m + s +1)+2 ⊗ w j +1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s − , then b ij = e j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 59 If s − j < s , then b ij = − e j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +2) ) f ( j, s − w j + m + s − f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1) ,i = ( j + 1) s ; κ ℓ ( α j + m +2) ) f ( j, s − e j + m +1)+3 ⊗ w j +3 → j + s − f ( j, s − s +1)+1 ,i = 2 s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (6) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − w j + m ) − → j + m )+2 ⊗ e j , i = j ; w j + m )+1 → j + m )+2 ⊗ w j → j + s )+1 , i = j + 3 s ; − e j + m )+2 ⊗ w j → j + s )+2 , i = j + 7 s ;0 , otherwise.If s j < s , then b ij = w j + m )+1 → j + m )+2 ⊗ w j → j + s )+1 , i = j + s ; − e j + m )+2 ⊗ w j → j + s )+2 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +1) ) e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − or j = 4 s − , otherwise.If s j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (6) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +1) ) w j + m )+1 → j + m )+3 ⊗ e j , i = j ; κ ℓ ( α j + m +1) ) w j + m + s )+1 → j + m )+3 ⊗ e j , i = j + s ; − κ ℓ ( α j + m +1) ) e j + m )+3 ⊗ w j → j +2 , i = j + 4 s ; − κ ℓ ( α j + m +1) ) e j + m )+3 ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 . If s j < s , then b ij = κ ℓ ( α j + m +3) ) f ( j, s − e j + m +2) ⊗ w j +3 → j + s − f ( j, s − s +1)+1 ,i = 2 s + ( j + 1) s ;0 , otherwise. (7) If r = 6 , then Ω ( Y (6) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ ( α j + m +1) ) w j + m ) → j + m )+3 ⊗ e j , i = j ; κ ℓ ( α j + m +1) ) w j + m )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ; κ ℓ ( α j + m +1) ) w j + m + s )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +2) ) e j + m +1) ⊗ w j + s )+1 → j +1) ,i = ( j + 1) s , j < s − or j = 3 s − , otherwise.If s j < s , then b ij = − w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +1) ,i = ( j + 1) s , j < s − or j = 5 s − , otherwise.If s j < s , then b ij = f ( j, s − w j + m +1) → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = 3 s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = − e j + m + s +1)+2 ⊗ w j +3 → j +1)+1 , i = 4 s + ( j + 1) s ; e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+2 , i = 5 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 61 If s j < s , then b ij = − f ( j, s − w j + m +1) → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = 3 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = − e j + m + s +1)+2 ⊗ w j +3 → j +1)+1 , i = 4 s + ( j + 1) s ; e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+2 , i = 5 s + ( j + 1) s ;0 , otherwise. (8) If r = 7 , then Ω ( Y (6) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m + s )+2 → j + m +1) ⊗ e j , i = j + s ; − κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ; − κ ℓ ( α j + m +2) ) e j + m +1) ⊗ w j → j +2 , i = j + 4 s ; − κ ℓ ( α j + m +2) ) e j + m +1) ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = e j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m + s − f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise. (9) If r = 8 , then Ω ( Y (6) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1) ⊗ w j → j +1) , i = ( j + 1) s ; κ ℓ ( α j + m +2) ) w j + m )+2 → j + m +1) ⊗ w j → j +1 , i = j + s ; κ ℓ ( α j + m +2) ) w j + m + s )+2 → j + m +1) ⊗ w j → j + s )+1 , i = j + 2 s ; κ ℓ ( α j + m +2) ) e j + m +1) ⊗ w j → j +3 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s , j < s − or j = 3 s − , otherwise.If s j < s − , then b ij = w j + m )+3 → j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 , i = 4 s + ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m )+3 → j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 , i = 4 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 63 If s j < s , then b ij = − κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; − κ ℓ ( α j + m +2) ) w j + m +1+ f ( j, s − s )+2 → j + m +1)+3 ⊗ w j +3 → j +1+ f ( j, s − s )+1 ,i = 2 s + ( j + 1) s ; − κ ℓ ( α j + m +2) ) w j + m +1+ f ( j, s − s )+1 → j + m +1)+3 ⊗ w j +3 → j + s − f ( j, s − s +1)+2 ,i = 3 s + ( j + 1) s ;0 , otherwise. (10) If r = 9 , then Ω ( Y (6) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ e j , i = j ; w j + m +1) → j + m + s +1)+1 ⊗ w j → j +1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j → j + s )+1 , i = j ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s − , then b ij = − f ( j, s − κ ℓ ( α j + m +2) ) e j + m +1)+3 ⊗ w j +2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = f ( j, s − κ ℓ ( α j + m +3) ) e j + m +2) ⊗ w j +3 → j +1+ f ( j, s − s )+1 , i = s + ( j + 1) s ;0 , otherwise. (11) If r = 10 , then Ω ( Y (6) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +1) ) w j + m ) → j + m )+3 ⊗ e j , i = j ; − κ ℓ ( α j + m +1) ) w j + m + s )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ; κ ℓ ( α j + m +1) ) w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 2 s ; κ ℓ ( α j + m +1) ) w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 , i = j + 3 s ; − κ ℓ ( α j + m +1) ) w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 , i = j + 4 s ; κ ℓ ( α j + m +1) ) e j + m )+3 ⊗ w j → j +3 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = f ( j, s − w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j + s +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j + s +1)+1 , i = 2 s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w j + m )+2 → j + m +1)+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ; − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (6) t ) is a Ω r ( Y (6) t ) , whose left components twisted by σ ℓ . Proposition 23 (Translates for the case 7) . (I) Let r ∈ N , r < . r -translates of the elements Y (7) t are described by the following way. (1) If r = 0 , then Ω ( Y (7) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j → j + s )+2 ⊗ e j , i = ( j ) s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 65 If s j < s , then b ij = w j +1 → j +3 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w j +2 → j +1) ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (2) If r = 1 , then Ω ( Y (7) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+1 → j + m )+3 ⊗ e j , i = j ; w j + m + s )+1 → j + m )+3 ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m )+1 → j + m +1) ⊗ e j , i = j ; w j + m + s )+2 → j + m +1) ⊗ w j → j + s )+1 , i = j + s ; w j + m )+2 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = w j + m )+2 → j + m +1)+1 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = w j + m )+2 → j + m +1)+1 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+2 , i = j − s ; w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 . If s j < s , then b ij = w j + m )+3 → j + m +1)+2 ⊗ e j +2 , i = j − s ; w j + m +1) → j + m +1)+2 ⊗ w j +2 → j +3 , i = j − s ;0 , otherwise.If s j < s , then b ij = w j + m +1+ f ( j, s − s )+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise. (3) If r = 2 , then Ω ( Y (7) t ) is described with (8 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m )+2 → j + m +1) ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w j + m )+2 → j + m +1) ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = 2 s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 67 If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = s + f ( j, s − s + ( j + 1) s ;0 , otherwise. (4) If r = 3 , then Ω ( Y (7) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+2 → j + m +1) ⊗ e j , i = j ; w j + m + s )+2 → j + m +1) ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + m +1)+2 ⊗ w j +1 → j +1) , i = f ( j, s − s + ( j + 1) s ; w j + m )+3 → j + m +1)+2 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = s − f ( j, s − s + ( j + 1) s ; w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m + s − f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m − f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (7) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m + s ) − → j + m + s )+2 ⊗ e j , i = j ; w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +1 , i = j + 2 s ; e j + m + s )+2 ⊗ w j → j +2 , i = j + 5 s ;0 , otherwise. If s j < s , then b ij = w j + m + s ) − → j + m + s )+2 ⊗ e j , i = j − s ; w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +1 , i = j + 2 s ; e j + m + s )+2 ⊗ w j → j +2 , i = j + 6 s ;0 , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m )+3 ⊗ e j +1 , i = j ; w j + m + s )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m )+3 ⊗ e j +1 , i = j ; w j + m + s )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (7) t ) is described with (6 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = s − f ( j, s − s + ( j + 1) s ; w j + m +1) → j + m +1)+2 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = f ( j, s − s + ( j + 1) s ; w j + m +1) → j + m +1)+2 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + m +2) ⊗ w j +3 → j + s − f ( j, s − s +1)+1 , i = 2 s + ( j + 1) s ; e j + m +2) ⊗ w j +3 → j +1+ f ( j, s − s )+1 , i = 3 s + ( j + 1) s ;0 , otherwise. (7) If r = 6 , then Ω ( Y (7) t ) is described with (7 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 69 If j < s , then b ij = w j + m )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ; w j + m + s )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = e j + m +1) ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m + s )+1 → j + m +1) ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m +1) ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m + s )+1 → j + m +1) ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+1 ⊗ w j + s )+2 → j + s +1)+1 , i = s + f ( j, s − s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+1 ⊗ w j + s )+2 → j + s +1)+1 , i = 3 s − f ( j, s − s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j +1)+1 , i = 3 s − f ( j, s − s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j +1)+1 , i = s + f ( j, s − s + ( j + 1) s ;0 , otherwise. (8) If r = 7 , then Ω ( Y (7) t ) is described with (6 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j + s ; w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ; e j + m + s +1)+1 ⊗ w j + s )+1 → j +3 , i = j + 5 s ;0 , otherwise. If s j < s , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j + s ; w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ; e j + m + s +1)+1 ⊗ w j + s )+1 → j +3 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 . (9) If r = 8 , then Ω ( Y (7) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+3 → j + m +1) ⊗ w j → j +1) , i = ( j + 1) s ; w j + m )+2 → j + m +1) ⊗ w j → j +1 , i = j + s ; w j + m + s )+2 → j + m +1) ⊗ w j → j + s )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m + s )+2 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m + s )+2 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j + s )+2 → j + s +1)+1 , i = s + f ( j, s − s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j + s )+2 → j + s +1)+1 , i = 2 s − f ( j, s − s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w j + m + s − f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j + s − f ( j, s − s +1)+1 , i = s + ( j + 1) s ; w j + m + f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j + f ( j, s − s +1)+1 , i = 2 s + ( j + 1) s ;0 , otherwise. (10) If r = 9 , then Ω ( Y (7) t ) is described with (7 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 71 If s j < s , then b ij = w j + m +1) → j + m +1)+2 ⊗ e j +1 , i = j − s ; e j + m +1)+2 ⊗ w j +1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m +1)+2 ⊗ e j +1 , i = j − s ; e j + m +1)+2 ⊗ w j +1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (7) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m + s )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ; w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 2 s ; w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 , i = j + 3 s ; w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m +1) → j + m +2) ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m + s − f ( j, s − s +1)+1 → j + m +2) ⊗ w j +3 → j + f ( j, s − s +1)+1 , i = 2 s + ( j + 1) s ; w j + m + s − f ( j, s − s +1)+2 → j + m +2) ⊗ w j +3 → j + f ( j, s − s +1)+2 , i = 4 s + ( j + 1) s ;0 , otherwise. (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (7) t ) is a Ω r ( Y (7) t ) , whose left components twisted by σ ℓ . Proposition 24 (Translates for the case 8) . (I) Let r ∈ N , r < . r -translates of the elements Y (8) t are described by the following way. (1) If r = 0 , then Ω ( Y (8) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b , = w j + m ) → j + m )+3 ⊗ e j . (2) If r = 1 , then Ω ( Y (8) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b s +( j +1) s , s − = w j + m + s +1)+1 → j + m +2) ⊗ w j +1 → j +1) ; b s +( j +1) s , s − = w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +2 → j +1) . (3) If r = 2 , then Ω ( Y (8) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , s − = w j + m )+3 → j + m +1+ s )+1 ⊗ w j +1 → j +1) . (4) If r = 3 , then Ω ( Y (8) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b s +( j +1) s , s − = e j + m + s +1)+2 ⊗ w j → j +1) ; b s +( j +1) s , s − = w j + m + s +1)+2 → j + m +1)+3 ⊗ w j +1 → j +1) . (5) If r = 4 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s , s − = w j + m )+3 → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s − = w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j +1) . (6) If r = 5 , then Ω ( Y (8) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s ,s − = w j + m +1+ sf ( j,s − → j + m +1)+3 ⊗ w j → j +1) ; b ( j +1) s , s − = w j + m + s +1)+1 → j + m +2) ⊗ w j + s )+1 → j +1) . (7) If r = 6 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s , s − = w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s − = e j + m + s +1)+1 ⊗ w j + s )+1 → j + s +1)+1 . (8) If r = 7 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s ,s − = w j + m +1+ sf ( j,s − → j + m +2) ⊗ w j → j +1) ; b ( j +1) s , s − = w j + m + s +1)+2 → j + m + s +2)+1 ⊗ w j + s )+1 → j +1) . (9) If r = 8 , then Ω ( Y (8) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s , s − = w j + m )+3 → j + m +1)+2 ⊗ w j +1 → j +1) ; b s +( j +1) s , s − = e j + m +1)+2 ⊗ w j +1 → j +1)+1 . (10) If r = 9 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b j + s,s − = w j + m +1) → j + m +1)+3 ⊗ w j → j +1 ; b j +5 s,s − = w j + m +1)+2 → j + m )+3 ⊗ w j → j +3 , s = 1; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 73 b j +2 s,s − = w j + m +1) → j + m +1)+3 ⊗ w j → j + s )+1 , s > b ( j +1) s , s − = w j + m +1)+3 → j + m + s +2)+2 ⊗ w j + s )+1 → j +1) . (11) If r = 10 and s = 1 , then Ω ( Y (8) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j, = w j + m ) → j + m +1)+2 ⊗ e j ; b j +5 s, = w j + m )+3 → j + m +1)+2 ⊗ w j → j +3 ; b j − s,s = w j + m ) → j + m +1)+2 ⊗ e j ; b j − s, s = w j + m ) → j + m )+3 ⊗ w j +1 → j ; b j − s, s = e j + m ) ⊗ w j +2 → j ; b j − s, s = w j + m )+3 → j + m )+1 ⊗ e j +3 . (12) If r = 10 and s > , then Ω ( Y (8) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j +2 s, s − = w j + m )+2 → j + m +1)+2 ⊗ w j → j + s )+2 ; b j +4 s, s − = w j + m )+3 → j + m +1)+2 ⊗ w j → j +3 ; b s +( j +1) s , s − = w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1)+1 ; b ( j +1) s , s − = w j + m +1) → j + m +1)+3 ⊗ w j +1 → j +1) ; b j +2 s, s − = w j + m )+3 → j + m +1)+3 ⊗ w j +1 → j +3 ; b j +2 s, s − = w j + m )+3 → j + m +1)+3 ⊗ w j +1 → j +3 ; b s +( j +1) s , s − = w j + m + s +1)+1 → j + m +2) ⊗ w j +2 → j +1)+1 ; b s +( j +1) s , s − = w j + m + s +1)+2 → j + m +2) ⊗ w j +2 → j +1)+2 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (8) t ) is a Ω r ( Y (8) t ) , whose left components twisted by σ ℓ . Proposition 25 (Translates for the case 9) . (I) Let r ∈ N , r < . r -translates of the elements Y (9) t are described by the following way. (1) If r = 0 , then Ω ( Y (9) t ) is described with (9 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = κ ℓ ( α j + m +2) ) e j ⊗ e j , i = j − s ;0 , otherwise.If s j < s , then b ij = e j +1 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + s )+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . If s j < s , then b ij = e j +2 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) e j +3 ⊗ e j +3 , i = j − s ;0 , otherwise. (2) If r = 1 , then Ω ( Y (9) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m + s )+1 ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j , i = j − s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +3) ) w j + m )+2 → j + m +1) ⊗ e j +1 , i = j ; − κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1) ⊗ w j +1 → j +2 , i = j + 2 s ; − κ ℓ ( α j + m +3) ) e j + m +1) ⊗ w j +1 → j +3 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +3) ) w j + m )+2 → j + m +1) ⊗ e j +1 , i = j ; − κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1) ⊗ w j +1 → j +2 , i = j + 2 s ; − κ ℓ ( α j + m +3) ) e j + m +1) ⊗ w j +1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ e j +2 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 75 If s j < s , then b ij = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + s + 1) s ; e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = 2 s + ( j + s + 1) s ; − w j + m +1) → j + m + s +1)+2 ⊗ e j +3 , i = j ;0 , otherwise.If s j < s , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + s + 1) s ; − e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = 2 s + ( j + s + 1) s ; w j + m +1) → j + m + s +1)+2 ⊗ e j +3 , i = j − s ;0 , otherwise. (3) If r = 2 , then Ω ( Y (9) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m ) − → j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = − e j + m )+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e j + m )+2 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = − e j + m + s )+1 ⊗ w j +1 → j +2 , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = − w j + m )+1 → j + m )+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise. If s j < s , then b ij = − w j + m )+1 → j + m )+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +2) ) f ( j, s − e j + m )+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1) ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m +3) ) e j + m +1) ⊗ e j +3 , i = j − s ;0 , otherwise. (4) If r = 3 , then Ω ( Y (9) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − e j + m + s )+2 ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j , i = j − s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +3) ) e j + m +1) ⊗ e j +2 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 77 If s j < s , then b ij = − κ ℓ ( α j + m +3) ) e j + m +1) ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+1 ⊗ e j +3 , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+1 ⊗ e j +3 , i = j − s ;0 , otherwise. (5) If r = 4 , then Ω ( Y (9) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ ( α j + m +1) ) e j +2) − ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = − w j + m )+1 → j + m )+2 ⊗ e j + s )+1 , i = j + s ; − e j + m )+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − w j + m )+1 → j + m )+2 ⊗ e j + s )+1 , i = j + s ; − e j + m )+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j + s )+2 , i = j + 2 s ;0 , otherwise. If s j < s , then b ij = − κ ℓ ( α j + m +3) ) f ( j, s − e j + m +1) ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; − κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1) ⊗ e j +3 , i = j + 3 s ;0 , otherwise. (6) If r = 5 , then Ω ( Y (9) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m )+1 → j + m )+3 ⊗ e j , i = j ; κ ℓ ( α j + m +2) ) w j + m + s )+1 → j + m )+3 ⊗ e j , i = j + s ; κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ w j → j +2 , i = j + 4 s ; κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +3) ) e j + m +1) ⊗ e j + s )+1 , i = j + s ; κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +3) ) e j + m +1) ⊗ e j + s )+1 , i = j + s ; κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − e j + m + s +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + s + 1) s ; − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − e j + m + s +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + s + 1) s ; − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s ; − e j + m +1)+2 ⊗ e j +3 , i = j + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 79 If s j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s ; e j + m +1)+2 ⊗ e j +3 , i = j ;0 , otherwise. (7) If r = 6 , then Ω ( Y (9) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ ( α j + m +2) ) e j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ e j +3 , i = j + 2 s ;0 , otherwise. (8) If r = 7 , then Ω ( Y (9) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +3) ) w j + m + s )+2 → j + m +1) ⊗ e j , i = j + s ; − κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ; κ ℓ ( α j + m +3) ) e j + m +1) ⊗ w j → j +2 , i = j + 4 s ; κ ℓ ( α j + m +3) ) e j + m +1) ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise. If s j < s , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j + s ; w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ; − e j + m + s +1)+1 ⊗ w j + s )+1 → j +3 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j + s ; w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ; − e j + m + s +1)+1 ⊗ w j + s )+1 → j +3 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = − e j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + s + 1) s ; − w j + m +1) → j + m + s +1)+2 ⊗ e j + s )+2 , i = j + s ; w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − e j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + s + 1) s ; − w j + m +1) → j + m + s +1)+2 ⊗ e j + s )+2 , i = j + s ; w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +3) ) w j + m + s − f ( j, s − s +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; − κ ℓ ( α j + m +3) ) e j + m +1)+3 ⊗ w j +3 → j + f ( j, s − s +1)+1 , i = 2 s + ( j + 1) s ; − κ ℓ ( α j + m +3) ) w j + m + s +1)+1 → j + m +1)+3 ⊗ e j +3 , i = j + s ; − κ ℓ ( α j + m +3) ) w j + m +1)+1 → j + m +1)+3 ⊗ e j +3 , i = j + 2 s ;0 , otherwise. (9) If r = 8 , then Ω ( Y (9) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m )+1 ⊗ w j → j + s )+2 , i = j + 4 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 81 If s j < s , then b ij = − w j + m ) − → j + m )+1 ⊗ e j , i = j − s ; e j + m )+1 ⊗ w j → j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = − e j + m )+2 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = − e j + m )+2 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s − , then b ij = − κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ w j +2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m +2) ) w j + m + s )+1 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ w j +2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +2) ) w j + m + s )+1 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +3) ) f ( j, s − e j + m +1) ⊗ e j +3 , i = j − s ;0 , otherwise. (10) If r = 9 , then Ω ( Y (9) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ e j , i = j ;0 , otherwise. If s j < s , then b ij = − w j + m +1) → j + m + s +1)+2 ⊗ e j + s )+1 , i = j ; − e j + m + s +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − w j + m +1) → j + m + s +1)+2 ⊗ e j + s )+1 , i = j ; − e j + m + s +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − e j + m +1)+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = − e j + m +1)+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − κ ℓ ( α j + m + s − ) w j + m +1)+3 → j + m +2) ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m + s − ) e j + m +2) ⊗ w j +3 → j + f ( j, s − s +1)+1 , i = s + ( j + 1) s ; κ ℓ ( α j + m + s − ) w j + m + s +1)+2 → j + m +2) ⊗ e j +3 , i = j ;0 , otherwise. (11) If r = 10 , then Ω ( Y (9) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) → j + m )+2 ⊗ e j , i = j ; w j + m )+1 → j + m )+2 ⊗ w j → j + s )+1 , i = j + 2 s ; e j + m )+2 ⊗ w j → j + s )+2 , i = j + 4 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 83 If s j < s , then b ij = w j + m ) → j + m )+2 ⊗ e j , i = j − s ; − w j + m )+1 → j + m )+2 ⊗ w j → j + s )+1 , i = j ; − e j + m )+2 ⊗ w j → j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +2) ) f ( j, s − w j + m + s )+1 → j + m )+3 ⊗ e j +1 , i = j − s ; κ ℓ ( α j + m +2) ) f ( j, s − w j + m + s )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + s ; κ ℓ ( α j + m +2) ) f ( j, s − e j + m )+3 ⊗ w j +1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) f ( j, s − w j + m + s )+1 → j + m )+3 ⊗ e j +1 , i = j − s ; − κ ℓ ( α j + m +2) ) f ( j, s − w j + m + s )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + s ; κ ℓ ( α j + m +2) ) f ( j, s − e j + m )+3 ⊗ w j +1 → j +3 , i = j + 2 s ;0 , otherwise.If s j < s − , then b ij = κ ℓ ( α j + m + s − ) e j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m + s − ) w j + m + s )+2 → j + m +1) ⊗ e j +2 , i = j − s ; κ ℓ ( α j + m + s − ) w j + m )+3 → j + m +1) ⊗ w j +2 → j +3 , i = j + s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − κ ℓ ( α j + m + s − ) e j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m + s − ) w j + m + s )+2 → j + m +1) ⊗ e j +2 , i = j − s ;0 , otherwise. If s j < s , then b ij = e j + m +1)+1 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + s + 1) s ; w j + m )+3 → j + m +1)+1 ⊗ e j +3 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s − , then b ij = w j + m +1) → j + m +1)+1 ⊗ w j +3 → j + s − f ( j, s − s +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1) → j + m +1)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + m +1)+1 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + s + 1) s ;0 , otherwise. (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (9) t ) is a Ω r ( Y (9) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 26 (Translates for the case 10) . (I) Let r ∈ N , r < . Denote by κ = κ ℓ ( α ) , s = 1; − κ ℓ ( γ ) κ ℓ ( α ) , s > . Then r -translates of the elements Y (10) t are described by the following way. (1) If r = 0 , then Ω ( Y (10) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b ,s = κ ℓ ( α ) w j → j +1) ⊗ e j . (2) If r = 1 , then Ω ( Y (10) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b ( j + s +1) s , s = f ( s, κ w j + m + s +1)+1 → j + m + s +2) ⊗ w j +1 → j +1) ; b ( j + s +1) s , s = − f ( s, κ w j + m + s +1)+1 → j + m + s +2) ⊗ w j +1 → j +1) . (3) If r = 2 , then Ω ( Y (10) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , = κ w j + m )+3 → j + m +1) ⊗ w j → j +1) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 85 (4) If r = 3 , then Ω ( Y (10) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s , = − κ ℓ ( α j + m +2) ) κ e j + m +1)+2 ⊗ w j → j +1) ; b ( j +1) s ,s = − κ ℓ ( α j + m +2) ) κ e j + m +1)+2 ⊗ w j → j +1) . (5) If r = 4 , then Ω ( Y (10) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , = − f ( s, κ e j + m )+3 ⊗ w j → j +1) . (6) If r = 5 , then Ω ( Y (10) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b ( j + s +1) s , = − f ( s, κ ℓ ( γ j + m ) κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j → j +1) ; b ( j +1) s , = − f ( s, κ ℓ ( γ j + m ) κ w j + m +1)+1 → j + m +1)+3 ⊗ w j → j +1) . (7) If r = 6 , then Ω ( Y (10) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , = − f ( s, κ e j + m +1) ⊗ w j → j +1) . (8) If r = 7 , then Ω ( Y (10) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , = κ κ w j + m + sf ( j,s − → j + m +2) ⊗ w j → j +1) , where κ = − κ ℓ ( γ m ) , s = 1; κ ℓ ( γ m ) κ ℓ ( γ m +1 ) κ ℓ ( γ m +2 ) , s > . (9) If r = 8 , then Ω ( Y (10) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s ,s (1 − f ( s, = κ κ w j + m )+3 → j + m + s +1)+1 ⊗ w j → j +1) , where κ = − κ ℓ ( α m +4) ) , s = 1; κ ℓ ( γ m − ) κ ℓ ( γ m ) κ ℓ ( γ m +1 ) κ ℓ +1 ( α ) , s > . (10) If r = 9 and s = 1 , then Ω ( Y (10) t ) is described with (6 s × s ) -matrix with one nonzeroelement that is of the following form : b ( j +1) s , = − κ ℓ ( γ j + m ) κ e j + m +1)+3 ⊗ w j → j +1) . (11) If r = 9 and s > , then Ω ( Y (10) t ) is described with (6 s × s ) -matrix with the followingtwo nonzero elements : b s + j, = − κ κ w j + m +1)+2 → j + m +1)+3 ⊗ w j → j +3 ; b s + j, = κ κ w j + m + s +1)+2 → j + m +1)+3 ⊗ w j → j +3 , where κ = − κ ℓ ( γ m − ) κ ℓ ( γ m ) κ ℓ ( γ m +1 ) . (12) If r = 10 and s = 1 , then Ω ( Y (10) t ) is described with (8 s × s ) -matrix with the followingtwo nonzero elements : b ( j +1) s , = − κ ℓ ( α j + m +3) ) κ w j + m +1) → j + m + s +1)+2 ⊗ w j → j +1) ; b ( j +1) s ,s = κ ℓ ( α j + m +3) ) κ w j + m +1) → j + m + s +1)+2 ⊗ w j → j +1) . (13) If r = 10 and s > , then Ω ( Y (10) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b ( j +1) s , = κ κ w j + m +1) → j + m + s +1)+2 ⊗ w j → j +1) ; b s + j, = − κ κ w j + m +1) − → j + m + s +1)+2 ⊗ w j → j +1) − ; b ( j +1) s ,s = κ κ w j + m +1) → j + m + s +1)+2 ⊗ w j → j +1) ; b s + j,s = − κ κ w j + m +1) − → j + m + s +1)+2 ⊗ w j → j +1) − ; b s +( j +1) s , s − sf (( j ) s , = κ κ w j + m + s +1)+3 → j + m + s +1)+5 ⊗ w j +3 → j +1)+3 , where κ = κ ℓ ( γ m − ) κ ℓ ( γ m − ) κ ℓ ( γ m ) κ ℓ ( γ m +1 ) κ ℓ +1 ( α ) . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (10) t ) is a Ω r ( Y (10) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 27 (Translates for the case 11) . (I) Let r ∈ N , r < . r -translates of theelements Y (11) t are described by the following way. (1) If r = 0 , then Ω ( Y (11) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) → j + m )+1 ⊗ e j , i = ( j ) s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +2+1) ) w j + m )+1 → j + m +1) ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +2) ) w j + m )+2 → j + m )+3 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) w j + m )+3 → j + m +1)+2 ⊗ e j +3 , i = 5 s + ( j ) s ;0 , otherwise. (2) If r = 1 , then Ω ( Y (11) t ) is described with (9 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 87 If j < s , then b ij = κ w j + m )+1 → j + m +1) ⊗ e j , i = j ; κ w j + m + s )+1 → j + m +1) ⊗ e j , i = j + s ; − κ w j + m )+2 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ; − κ w j + m + s )+2 → j + m +1) ⊗ w j → j + s )+1 , i = j + 3 s ;0 , otherwise,where κ = κ ℓ ( α j + m +3) ) .If s j < s , then b ij = w j + m + s )+2 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j + s ; w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ; w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+1 → j +3 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +1 → j +1) , i = ( j + s + 1) s ; − w j + m )+3 → j + m + s +1)+2 ⊗ w j +1 → j +2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = w j + m )+2 → j + m +1)+1 ⊗ e j +1 , i = j ; w j + m )+3 → j + m +1)+1 ⊗ w j +1 → j +2 , i = j + 2 s ; w j + m +1) → j + m +1)+1 ⊗ w j +1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+1 → j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise. If s j < s , then b ij = κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = 3 s + ( j + 1) s ; κ w j + m +1) → j + m +1)+3 ⊗ e j +3 , i = j − s ;0 , otherwise,where κ = − κ ℓ ( α j + m +3) ) .If s j < s , then b ij = κ w j + m + s +1)+1 → j + m +2) ⊗ w j +3 → j +1) , i = ( j + s + 1) s ; κ w j + m +1) → j + m +2) ⊗ e j +3 , i = j − s ;0 , otherwise,where κ = κ ℓ ( α j + m +4) ) . (3) If r = 2 , then Ω ( Y (11) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − f ( j, s ) w j + m ) − → j + m )+2 ⊗ e j , i = ( j ) s ; − e j + m )+2 ⊗ w j → j +1 , i = j + s ;0 , otherwise.If s j < s , then b ij = κ ℓ +1 ( α j + m +2) ) e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +3) ) w j + m + s )+1 → j + m +1) ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = − f (( j ) s , s − f ( j, s ) w j + m )+3 → j + m +1)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s ; e j + m +1)+1 ⊗ w j +3 → j + s +1)+2 , i = 3 s + ( j + s + 1) s ;0 , otherwise. (4) If r = 3 , then Ω ( Y (11) t ) is described with (6 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 89 If j < s , then b ij = − κ w j + m + s )+2 → j + m )+3 ⊗ e j , i = j + s ; κ e j + m )+3 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise,where κ = − κ ℓ ( α j + m +2) ) .If s j < s , then b ij = f (( j ) s , s − f ( j, s ) e j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + s + 1) s ;0 , otherwise.If s j < s , then b ij = − w j + m +1) → j + m +1)+1 ⊗ e j + s )+2 , i = j + s ; − e j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − w j + m +1) → j + m +1)+1 ⊗ e j + s )+2 , i = j + s ; − e j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κ w j + m + s +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j +1) , i = ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; κ w j + m +1)+3 → j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ;0 , otherwise,where κ = − κ ℓ ( α j + m +4) ) . (5) If r = 4 , then Ω ( Y (11) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m ) − → j + m )+3 ⊗ e j , i = j ; − κ e j + m )+3 ⊗ w j → j +1) , i = ( j + 1) s , j < s − − κ w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 3 s ; κ w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 , i = j + 5 s ;0 , otherwise,where κ = − κ ℓ ( α j + m +2) ) . If s j < s , then b ij = κ f (( j ) s , s − f ( j, s ) e j + m +1) ⊗ w j + s )+1 → j +1) , i = s + ( j + 1) s ; − κ w j + m )+1 → j + m +1) ⊗ e j + s )+1 , i = j + s ;0 , otherwise,where κ = − κ ℓ ( α j + m +3) ) .If s j < s , then b ij = w j + m )+3 → j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , j < s − − f (( j ) s , s − w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = s + ( j + 1) s ; − w j + m )+2 → j + m +1)+1 ⊗ e j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , j = 5 s − f ( j, s − w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = s + ( j + 1) s ; − w j + m )+2 → j + m +1)+1 ⊗ e j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1)+1 , i = 2 s + ( j + 1) s ; − w j + m )+3 → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s , j < s − e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 7 s + ( j + 1) s , j < s − e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 5 s + ( j + 1) s , j = 6 s − , otherwise.If s j < s , then b ij = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1)+1 , i = 2 s + ( j + 1) s ; e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 5 s + ( j + 1) s , j < s − − w j + m )+3 → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s , j = 7 s − e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 7 s + ( j + 1) s , j = 7 s − , otherwise. (6) If r = 5 , then Ω ( Y (11) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +3) ) e j + m +1) ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 91 If s j < s , then b ij = w j + m +1) → j + m +1)+1 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m +1)+2 ⊗ e j + s )+2 , i = j + s ; f ( j, s ) e j + m +1)+2 ⊗ w j + s )+2 → j +3 , i = 6 s + ( j ) s ;0 , otherwise.If s j < s , then b ij = − κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; − κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; κ e j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 4 s + ( j + 1) s ; κ e j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = 5 s + ( j + 1) s ;0 , otherwise,where κ = κ ℓ ( α j + m +3) ) f (( j ) s , s − . (7) If r = 6 , then Ω ( Y (11) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m )+1 → j + m +1) ⊗ w j → j +1 , i = j + s ; − κ w j + m + s )+2 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ; κ w j + m )+2 → j + m +1) ⊗ w j → j + s )+1 , i = j + 4 s ; κ e j + m +1) ⊗ w j → j +1) , i = ( j + 1) s , j = s − , otherwise,where κ = κ ℓ ( α j + m +3) ) .If s j < s , then b ij = w j + m )+2 → j + m +1)+1 ⊗ e j + s )+1 , i = j + 2 s − sf ( j, s ); − w j + m +1) → j + m +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s , s − j < s − , otherwise.If s j < s , then b ij = − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+2 → j +3 , i = 7 s + ( j ) s ; w j + m +1) → j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , s − j < s − , otherwise. If s j < s , then b ij = κ w j + m +1) → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = s + ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; − κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = 3 s + ( j + 1) s ; − κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = 4 s + ( j + 1) s ;0 , otherwise,where κ = − κ ℓ ( α j + m +3) ) f ( j, s − . (8) If r = 7 , then Ω ( Y (11) t ) is described with (7 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ w j → j + s )+1 , i = j + s ; w j + m +1) → j + m + s +1)+1 ⊗ w j → j + s )+2 , i = j + 3 s ; w j + m +1) → j + m + s +1)+1 ⊗ w j → j +2 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = κ w j + m + s +1)+2 → j + m +1)+3 ⊗ w j +2 → j +1) , i = ( j + s + 1) s ; κ w j + m +1) → j + m +1)+3 ⊗ e j +2 , i = j ; − κ w j + m +1)+1 → j + m +1)+3 ⊗ w j +2 → j +3 , i = j + 2 s ;0 , otherwise,where κ = κ ℓ ( α j + m +3) ) f ( j, s ) .If s j < s , then b ij = κ w j + m + s +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; − κ w j + m +1)+3 → j + m +2) ⊗ w j +3 → j +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; − κ e j + m +2) ⊗ w j +3 → j +1+ sf ( j, s − , i = 4 s + ( j + 1) s ; − κ e j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 5 s + ( j + 1) s ;0 , otherwise,where κ = − κ ℓ ( α j + m +4) ) . (9) If r = 8 , then Ω ( Y (11) t ) is described with (6 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 93 If j < s , then b ij = − κ w j + m ) − → j + m )+3 ⊗ e j , i = j ; κ f ( j, s − e j + m )+3 ⊗ w j → j +1) , i = ( j + 1) s ; κ w j + m )+2 → j + m )+3 ⊗ w j → j +1 , i = j + s ; − κ w j + m + s )+1 → j + m )+3 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise,where κ = κ ℓ ( α j + m +2) ) .If s j < s , then b ij = − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = 5 s + ( j ) s ;0 , otherwise.If s j < s , then b ij = κ w j + m + s +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = s + ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 3 s + ( j + 1) s ;0 , otherwise,where κ = κ ℓ ( α j + m − ) κ ℓ +1 ( γ j + m − ) . (10) If r = 9 , then Ω ( Y (11) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ e j , i = j ; − w j + m +1) → j + m + s +1)+2 ⊗ w j → j + s )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ w j → j + s − sf ( j,s ))+1 , i = j ;0 , otherwise.If s j < s , then b ij = κ w j + m +1) → j + m +1)+3 ⊗ e j +1 , i = j − s ; − κ e j + m +1)+3 ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − or j = 4 s − , otherwise, where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = κ w j + m + s +1)+1 → j + m +2) ⊗ e j +2 , i = j − s ; − κ w j + m +1)+3 → j + m +2) ⊗ w j +2 → j +1) , i = ( j + 1) s , s − j < s − , otherwise,where κ = f ( j, s ) κ ℓ ( α j + m − ) κ ℓ ( γ j + m − ) .If s j < s , then b ij = w j + m +1)+3 → j + m + s +2)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s , j < s − or j = 8 s − − w j + m +2) → j + m + s +2)+1 ⊗ w j +3 → j +1)+1 ,i = s + ( j + 1) s , j < s − or j = 8 s − − w j + m +2) → j + m + s +2)+1 ⊗ w j +3 → j + s +1)+1 ,i = s + ( j + 1) s , s − j < s − , otherwise. (11) If r = 10 , then Ω ( Y (11) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ w j + m ) → j + m )+3 ⊗ e j , i = j ; κ w j + m + s )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ; − κ w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 2 s ; − κ w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 , i = j + 3 s ; κ w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 , i = j + 4 s ; − κ e j + m )+3 ⊗ w j → j +3 , i = j + 5 s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = κ w j + m ) → j + m +1) ⊗ e j , i = j − s ; − κ w j + m + s )+1 → j + m +1) ⊗ w j → j +1 , i = j + s ; κ w j + m )+2 → j + m +1) ⊗ w j → j + s )+2 , i = j + 2 s ; − κ w j + m + s )+2 → j + m +1) ⊗ w j → j +2 , i = j + 3 s ; κ w j + m )+3 → j + m +1) ⊗ w j → j +3 , i = j + 4 s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 95 If s j < s − , then b ij = − w j + m +1) → j + m +1)+1 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m +1) → j + m +1)+1 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − w j + m + s )+2 → j + m + s +1)+1 ⊗ e j +2 , i = j − s ; − w j + m )+3 → j + m + s +1)+1 ⊗ w j +2 → j +3 , i = j + s ; − w j + m +1) → j + m + s +1)+1 ⊗ w j +2 → j +1) , i = ( j + 1) s , j < s − , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − w j + m )+2 → j + m +1)+1 ⊗ e j + s )+2 , i = j − s ; w j + m )+3 → j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = j − s ; − w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , j = 7 s − , otherwise.If s j < s − , then b ij = w j + m +1) → j + m +1)+2 ⊗ w j +2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = s + ( j + 1) s ; − κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; − κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = 3 s + ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 4 s + ( j + 1) s ; − κ e j + m +1)+3 ⊗ w j +3 → j +1)+3 , i = 5 s + ( j + 1) s ;0 , otherwise, where κ = − κ ℓ +1 ( α j + m − ) . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (11) t ) is a Ω r ( Y (11) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 28 (Translates for the case 12) . (I) Let r ∈ N , r < . r -translates of theelements Y (12) t are described by the following way. (1) If r = 0 , then Ω ( Y (12) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b s, s = κ ℓ ( α j + m +2) ) w j + m )+2 → j + m )+3 ⊗ e j +2 ; b s, s = κ ℓ ( α j + m +2) ) w j + m )+2 → j + m )+3 ⊗ e j +2 . (2) If r = 1 , then Ω ( Y (12) t ) is described with (9 s × s ) -matrix with the following nonzeroelements : b j, (3) s = − κ w j + m )+1 → j + m +1) ⊗ e j ; b j + s, (3) s = κ w j + m + s )+1 → j + m +1) ⊗ e j ; b j +2 s, (3) s = − κ w j + m )+2 → j + m +1) ⊗ w j → j +1 ; b j +3 s, (3) s = κ w j + m + s )+2 → j + m +1) ⊗ w j → j + s )+1 ; b j, s +(3) s = − w j + m )+2 → j + m +1)+2 ⊗ e j +1 ; b j − s, s +(3) s = − w j + m + s )+2 → j + m + s +1)+2 ⊗ e j + s )+1 ; b ( j +1) s + f ( s, , s +(3) s − (1) s = − κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) ; b s +( j +1) s − f ( s, , s +(3) s − (1) s = κ w j + m +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) ; b s +( j +1) s + f ( s, , s +(3) s − (1) s = − κ w j + m + s +1)+2 → j + m +1)+3 ⊗ w j +3 → j + s +1)+1 ; b s +( j +1) s − f ( s, , s +(3) s − (1) s = κ w j + m +1)+2 → j + m +1)+3 ⊗ w j +3 → j +1)+1 , where κ = κ ℓ ( α s + m +3) ) , κ = − κ ℓ ( α s +(3) s − (1) s + m ) ) . (3) If r = 2 , then Ω ( Y (12) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s , s +(2) s = − f ( s, κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ w j +1 → j +1) ; b ( j +1) s , s +(2) s = f ( s, κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ w j +1 → j +1) . (4) If r = 3 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the following two nonzeroelements : b s +( j +1) s − f ( s, ,s +(2) s = − f ( s, e j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b ( j +1) s + f ( s, , s +(2) s = f ( s, e j + m +1)+2 ⊗ w j + s )+1 → j +1) . (5) If r = 4 , then Ω ( Y (12) t ) is described with (7 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , (2) s = − κ ℓ ( α j + m +2) ) e j + m )+3 ⊗ w j → j +1) ; b ( j +1) s ,s +(2) s + f ( s, = − f ( s, κ ℓ ( α j + m +2) ) w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +1) ; b s +( j +1) s ,s +(2) s + f ( s, = f ( s, κ ℓ ( α j + m +2) ) e j + m +1) ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s +(2) s − f ( s, = − f ( s, κ ℓ ( α j + m +2) ) e j + m +1) ⊗ w j + s )+1 → j +1) . (6) If r = 5 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s + f ( s, ,s +(2) s = e j + m + s +1)+1 ⊗ w j + s )+1 → j +1) ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 97 b s +( j +1) s − f ( s, , s +(2) s = e j + m + s +1)+1 ⊗ w j + s )+1 → j +1) . (7) If r = 6 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , (2) s = − κ ℓ ( α j + m +3) ) e j + m +1) ⊗ w j → j +1) ; b j,s +(2) s = w j + m + s )+1 → j + m + s +1)+1 ⊗ e j + s )+1 ; b j + s, s +(2) s = w j + m + s )+1 → j + m + s +1)+1 ⊗ e j + s )+1 . (8) If r = 7 , then Ω ( Y (12) t ) is described with (7 s × s ) -matrix with the following nonzeroelements : b j +2 s, (2) s = − w j + m )+3 → j + m +1)+1 ⊗ w j → j +1 ; b j +4 s, (2) s = − w j + m +1) → j + m +1)+1 ⊗ w j → j +2 ; b j +5 s, (2) s = − w j + m +1) → j + m +1)+1 ⊗ w j → j + s )+2 ; b j,s +(2) s = w j + m )+2 → j + m +1)+1 ⊗ e j ; b j + s,s +(2) s = − w j + m )+3 → j + m +1)+1 ⊗ w j → j + s )+1 ; b j +3 s,s +(2) s = − w j + m +1) → j + m +1)+1 ⊗ w j → j + s )+2 ; b j +4 s,s +(2) s = − w j + m +1) → j + m +1)+1 ⊗ w j → j +2 . (9) If r = 8 and s = 2 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j,s +1 = − w j + m + s )+2 → j + m + s +1)+2 ⊗ e j +2)+1 ; b s +( j +1) s , s +1 = − e j + m + s +1)+2 ⊗ w j +2)+1 → j + s +1)+1 ; b s +( j +1) s , s +1 = − e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 ; b ( j +1) s , s +1 = − w j + m )+3 → j + m +1)+1 ⊗ w j + s )+2 → j +1) ; b s +( j +1) s , s +1 = − e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 ; b s +( j +1) s , s +1 = − w j + m + s +1)+2 → j + m ) ⊗ w j +3 → j + s +1)+1 ; b s +( j +1) s , s +1 = − w j + m + s +1)+1 → j + m ) ⊗ w j +3 → j +1)+2 . (10) If r = 8 and s = 2 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b s +( j +1) s ,s +1 = − e j + m + s +1)+2 ⊗ w j + s )+1 → j + s +1)+1 ; b j, s − f ( s, = − w j + m + s )+2 → j + m + s +1)+2 ⊗ e j + s )+1 ; b ( j +1) s , s +1 = − w j + m )+3 → j + m +1)+1 ⊗ w j + s )+2 → j +1) ; b s +( j +1) s , s +1 = − e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 ; b s +( j +1) s , s − f ( s, = − e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 ; b s +( j +1) s , s +(1) s = − f ( s, κ ℓ +1 ( α j + m − ) w j + m +1+ sf ( s, → j + m +2) ⊗ w j +3 → j +1+ sf ( s, ; b s +( j +1) s , s +(1) s = − f ( s, κ ℓ +1 ( α j + m − ) w j + m +1+ sf ( s, → j + m +2) ⊗ w j +3 → j + s +1+ sf ( s, . (11) If r = 9 and s = 1 , then Ω ( Y (12) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j + s, = w j + m ) → j + m +1)+2 ⊗ w j → j +1 ; b j +2 s, = − w j + m ) → j + m +1)+2 ⊗ w j → j +1)+1 ; b j − s, s = κ ℓ +1 ( α j + m ) ) e j + m )+3 ⊗ w j +1 → j ; b j − s, s = κ ℓ +1 ( α j + m ) ) w j + m )+3 → j + m ) ⊗ w j +2 → j ; b j − s, s = − w j + m )+3 → j + m )+1 ⊗ w j +3 → j . (12) If r = 9 and s = 2 , then Ω ( Y (12) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j + s, = w j + m +1) → j + m +1)+2 ⊗ w j → j +1 ; b j +2 s, = − w j + m +1) → j + m +1)+2 ⊗ w j → j +2)+1 ; b ( j +1) s , s +1 = e j + m +1)+3 ⊗ w j +1 → j +1) ; b ( j +1) s , s +1 = w j + m +1)+3 → j + m ) ⊗ w j +2 → j +1) ; b ( j +1) s , s +1 = − w j + m +1)+3 → j + m +2)+1 ⊗ w j +3 → j +1) . (13) If r = 9 and s > , then Ω ( Y (12) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j + f ( s, ,s +1 = w j + m +1) → j + m +1)+2 ⊗ w j → j + s )+1 ; b j + s + f ( s, ,s +1 = − w j + m +1) → j + m +1)+2 ⊗ w j → j +1 ; b j − s, s +1 = − κ ℓ +1 ( α j + m ) ) w j + m +1) → j + m +1)+3 ⊗ e j +1 ; b j +3 s − f ( s, , s +1 = κ ℓ +1 ( α j + m ) ) w j + m +1)+2 → j + m +1)+3 ⊗ w j +1 → j +3 ; b ( j +1) s , s +1 = − κ ℓ +1 ( α j + m − ) w j + m +1)+3 → j + m +2) ⊗ w j +2 → j +1) ; b ( j +1) s , s +1 = − w j + m +1)+3 → j + m +2)+1 ⊗ w j +3 → j +1) . (14) If r = 10 and s = 1 , then Ω ( Y (12) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b j + s, = − κ w j + m +1)+1 → j + m )+3 ⊗ w j → j +1 ; b j +2 s, = κ w j + m )+1 → j + m )+3 ⊗ w j → j +1)+1 ; b j +3 s, = κ w j + m +1)+2 → j + m )+3 ⊗ w j → j +2 ; b j +4 s, = − κ w j + m )+2 → j + m )+3 ⊗ w j → j +1)+2 ; b j +5 s, = κ e j + m )+3 ⊗ w j → j +3 ; b j,s = − κ w j + m )+1 → j + m ) ⊗ w j → j +1)+1 ; b j + s,s = κ w j + m +1)+1 → j + m ) ⊗ w j → j +1 ; b j +2 s,s = κ w j + m )+2 → j + m ) ⊗ w j → j +1)+2 ; b j +3 s,s = − κ w j + m +1)+2 → j + m ) ⊗ w j → j +2 ; b j +4 s,s = κ w j + m )+3 → j + m ) ⊗ w j → j +3 ; b j − s, s = w j + m +1)+1 → j + m )+1 ⊗ e j +1 ; b j + s, s = − w j + m +1)+2 → j + m )+1 ⊗ w j +1 → j +2 ; b j +3 s, s = − w j + m )+3 → j + m )+1 ⊗ w j +1 → j +3 ; b j − s, s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +1)+2 → j +1 ; b j − s, s = − w j + m )+2 → j + m +1)+2 ⊗ e j +1)+2 ; b j − s, s = e j + m +1)+2 ⊗ w j +1)+2 → j +2 ; b j, s = − w j + m )+3 → j + m +1)+2 ⊗ w j +1)+2 → j +3 ; b j − s, s = − w j + m +1)+2 → j + m )+2 ⊗ e j +2 ; b j − s, s = w j + m )+3 → j + m )+2 ⊗ w j +2 → j +3 ; b j − s, s = − κ ℓ +1 ( α j + m ) ) w j + m ) → j + m )+3 ⊗ w j +3 → j , where κ = − κ ℓ +1 ( α m ) , κ = − κ ℓ +1 ( α s + m ) ) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 99 (15) If r = 10 and s = 2 , then Ω ( Y (12) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b j + s, = w j + m +2)+1 → j + m )+3 ⊗ w j → j +1 ; b j +2 s, = − w j + m )+1 → j + m )+3 ⊗ w j → j +2)+1 ; b j +3 s, = − w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 ; b j +4 s, = w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 ; b j +5 s, = − e j + m )+3 ⊗ w j → j +3 ; b j,s +1 = w j + m )+1 → j + m +1) ⊗ w j → j +2)+1 ; b j + s,s +1 = − w j + m +2)+1 → j + m +1) ⊗ w j → j +1 ; b j +2 s,s +1 = − w j + m )+2 → j + m +1) ⊗ w j → j + s )+2 ; b j +3 s,s +1 = w j + m + s )+2 → j + m +1) ⊗ w j → j +2 ; b j +4 s,s +1 = − w j + m )+3 → j + m +1) ⊗ w j → j +3 ; b j − s, s +1 = w j + m +2)+1 → j + m + s +1)+1 ⊗ e j +1 ; b j + s, s +1 = − w j + m + s )+2 → j + m + s +1)+1 ⊗ w j +1 → j +2 ; b j +3 s, s +1 = − w j + m )+3 → j + m + s +1)+1 ⊗ w j +1 → j +3 ; b j − s, s +1 = − w j + m )+2 → j + m +1)+2 ⊗ e j + s )+2 ; b j, s +1 = − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+2 → j +3 ; b s +( j +1) s , s = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +2 → j +1)+1 ; b s +( j +1) s , s = e j + m + s +1)+2 ⊗ w j +2 → j +1)+2 ; b j − s, s − = − w j + m + s )+2 → j + m + s +1)+2 ⊗ e j +2 ; b j − s, s − = w j + m )+3 → j + m + s +1)+2 ⊗ w j +2 → j +3 ; b ( j +1) s , s = w j + m +1) → j + m +1)+3 ⊗ w j +3 → j +1) ; b j − s, s = − w j + m )+3 → j + m +1)+3 ⊗ e j +3 ; b j − s, s − = w j + m )+3 → j + m +1)+3 ⊗ e j +3 . (16) If r = 10 and s > , then Ω ( Y (12) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b j +2 s, = − κ w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 ; b j +4 s, = κ w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 ; b j,s +1 = − κ w j + m )+1 → j + m +1) ⊗ w j → j + s )+1 ; b j +2 s,s +1 = κ w j + m )+2 → j + m +1) ⊗ w j → j + s )+2 ; b j +4 s,s +1 = κ w j + m )+3 → j + m +1) ⊗ w j → j +3 ; b j − s, s +1 = w j + m + s )+1 → j + m + s +1)+1 ⊗ e j +1 ; b j + s, s +1 = − w j + m + s )+2 → j + m + s +1)+1 ⊗ w j +1 → j +2 ; b j +3 s, s +1 = − w j + m )+3 → j + m + s +1)+1 ⊗ w j +1 → j +3 ; b j − s, s +1 = w j + m + s )+1 → j + m + s +1)+1 ⊗ e j +1 ; b j + s, s +1 = − w j + m + s )+2 → j + m + s +1)+1 ⊗ w j +1 → j +2 ; b j +2 s, s +1 = w j + m )+3 → j + m + s +1)+1 ⊗ w j +1 → j +3 ; b s +( j +1) s , s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+1 ; b s +( j +1) s , s = e j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 ; b j − s, s +1 = − w j + m )+2 → j + m +1)+2 ⊗ e j + s )+2 ;
00 MARIYA KACHALOVA b j, s +1 = − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+2 → j +3 ; b j − s, s +1 = − w j + m + s )+2 → j + m + s +1)+2 ⊗ e j +2 ; b j − s, s +1 = w j + m )+3 → j + m + s +1)+2 ⊗ w j +2 → j +3 ; b ( j +1) s , s = − κ w j + m +1) → j + m +1)+3 ⊗ w j +3 → j +1) ; b s +( j +1) s , s = − κ w j + m +1)+1 → j + m +1)+3 ⊗ w j +3 → j + s +1)+1 ; b s +( j +1) s , s = κ w j + m +1)+2 → j + m +1)+3 ⊗ w j +3 → j + s +1)+2 ; b j − s, s = κ w j + m )+3 → j + m +1)+3 ⊗ e j +3 ; b j − s, s +1 = − κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1)+3 ⊗ e j +3 , where κ = κ ℓ +1 ( α m ) , κ = κ ℓ +1 ( α s + m − ) , κ = κ ℓ +1 ( α m ) . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (12) t ) is a Ω r ( Y (12) t ) , whose left components twisted by σ ℓ . Proposition 29 (Translates for the case 13) . (I) Let r ∈ N , r < . r -translates of theelements Y (13) t are described by the following way. (1) If r = 0 , then Ω ( Y (13) t ) is described with (9 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = e j + m + s )+1 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + m )+1 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m )+3 → j + m +1) ⊗ e j +3 , i = j − s ;0 , otherwise. (2) If r = 1 , then Ω ( Y (13) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m )+2 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 101 If s j < s , then b ij = w j + m )+2 → j + m )+3 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m +1) ⊗ e j +2 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m +1)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m +1) → j + m +1)+1 ⊗ e j +3 , i = 6 s + ( j ) s ;0 , otherwise. (3) If r = 2 , then Ω ( Y (13) t ) is described with (6 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = e j + m + s )+2 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m +1) ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise. (4) If r = 3 , then Ω ( Y (13) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+2 → j + m )+3 ⊗ e j , i = j ; w j + m + s )+2 → j + m )+3 ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m +1) ⊗ e j + s )+1 , i = j + s ;0 , otherwise.
02 MARIYA KACHALOVA If s j < s , then b ij = w j + m +1) → j + m +1)+1 ⊗ e j + s )+2 , i = j + s ; e j + m +1)+1 ⊗ w j + s )+2 → j +3 , i = 6 s + ( j ) s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + s + 1) s ;0 , otherwise. (5) If r = 4 , then Ω ( Y (13) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) − → j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = e j + m )+1 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m )+2 ⊗ e j + s )+2 , i = j + 3 s − sf ( j, s );0 , otherwise.If s j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (13) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m +1) ⊗ w j → j +1 , i = j + 2 s ; e j + m +1) ⊗ w j → j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = e j + m +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m +1) → j + m +1)+1 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m +1)+2 ⊗ e j + s )+2 , i = j + s ; e j + m +1)+2 ⊗ w j + s )+2 → j +3 , i = 6 s + ( j ) s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 103 If s j < s , then b ij = w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; e j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 4 s + ( j + 1) s ; e j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = 5 s + ( j + 1) s ;0 , otherwise. (7) If r = 6 , then Ω ( Y (13) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m + s )+1 ⊗ w j → j + s )+1 , i = j + 3 sf ( j, s );0 , otherwise.If s j < s , then b ij = e j + m + s )+2 ⊗ e j +1 , i = j + s (1 − f ( j, s ));0 , otherwise.If s j < s , then b ij = w j + m )+2 → j + m )+3 ⊗ e j +2 , i = j + s ; e j + m )+3 ⊗ w j +2 → j +3 , i = 7 s + ( j ) s ;0 , otherwise.If s j < s , then b ij = e j + m +1) ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise. (8) If r = 7 , then Ω ( Y (13) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+2 → j + m )+3 ⊗ e j , i = j ; w j + m + s )+2 → j + m )+3 ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = e j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m +1)+2 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.
04 MARIYA KACHALOVA If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ; e j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m +1+ sf ( j, s − → j + m +2) ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; e j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 4 s + ( j + 1) s ; e j + m +2) ⊗ w j +3 → j +1+ sf ( j, s − , i = 5 s + ( j + 1) s ;0 , otherwise. (9) If r = 8 , then Ω ( Y (13) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) − → j + m + s )+2 ⊗ e j , i = ( j ) s ; e j + m + s )+2 ⊗ w j → j + s )+1 , i = s + ( j + s ) s ;0 , otherwise.If s j < s , then b ij = w j + m )+2 → j + m )+3 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w j + m )+3 → j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s , j < s − or j = 6 s − w j + m + s )+1 → j + m +1) ⊗ e j +2 , i = j − s ; e j + m +1) ⊗ w j +2 → j +3 , i = 5 s + ( j ) s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+1 ⊗ w j +3 → j +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise. (10) If r = 9 , then Ω ( Y (13) t ) is described with (9 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m )+3 → j + m +1) ⊗ e j , i = j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 105 If s j < s , then b ij = w j + m +1) → j + m +1)+1 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+1 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j + s )+2 → j +3 , i = j ;0 , otherwise.If s j < s , then b ij = e j + m +1)+1 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If s j < s , then b ij = e j + m +1)+2 ⊗ w j +2 → j +3 , i = j − s ;0 , otherwise.If s j < s , then b ij = e j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise. (11) If r = 10 , then Ω ( Y (13) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) → j + m + s )+1 ⊗ e j , i = ( j ) s ; e j + m + s )+1 ⊗ w j → j +1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m +1) ⊗ e j +1 , i = j − s ; w j + m + s )+2 → j + m +1) ⊗ w j +1 → j +2 , i = j + s ; w j + m )+3 → j + m +1) ⊗ w j +1 → j +3 , i = 5 s + ( j ) s ;0 , otherwise.
06 MARIYA KACHALOVA If s j < s , then b ij = w j + m + s )+2 → j + m )+3 ⊗ e j +2 , i = j − s ; e j + m )+3 ⊗ w j +2 → j +3 , i = 5 s + ( j ) s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s ; w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1)+1 , i = s + ( j + 1) s ; e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 3 s + ( j + 1) s , j < s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j +3 → j +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise. (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (13) t ) is a Ω r ( Y (13) t ) , whose left components twisted by σ ℓ . Proposition 30 (Translates for the case 14) . (I) Let r ∈ N , r < . r -translates of theelements Y (14) t are described by the following way. (1) If r = 0 , then Ω ( Y (14) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b s, s = κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1)+3 ⊗ e j +3 . (2) If r = 1 , then Ω ( Y (14) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j +2 s, s +(3) s = − κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1)+3 ⊗ w j +1 → j +2 ; b j +2 s, s +(3) s = − κ ℓ ( α j + m +3) ) w j + m )+3 → j + m +1)+3 ⊗ w j +1 → j +2 . (3) If r = 2 , then Ω ( Y (14) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, (3) s = − f ( s, κ ℓ ( α j + m +3) ) w j + m ) − → j + m )+3 ⊗ e j . (4) If r = 3 , then Ω ( Y (14) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , (2) s = − f ( s, κ ℓ ( α j + m +3) ) w j + m +1+ sf ( s, → j + m +1)+3 ⊗ w j → j +1) . (5) If r = 4 , then Ω ( Y (14) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s +( j +1) s , (2) s = − f ( s, κ ℓ ( α j + m +3) ) e j + m +1) ⊗ w j → j +1) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 107 (6) If r = 5 , then Ω ( Y (14) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , (2) s = κ ℓ +1 ( α j + m − ) w j + m +1+ sf ( s, → j + m +2) ⊗ w j → j +1) . (7) If r = 6 , then Ω ( Y (14) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s ,s +2 − f ( s, = − w j + m +1) → j + m + s +1)+1 ⊗ w j → j +1) . (8) If r = 7 , then Ω ( Y (14) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , (2) s = − κ ℓ +1 ( α j + m − ) w j + m +1+ sf ( s, → j + m +1)+3 ⊗ w j → j +1) . (9) If r = 8 , then Ω ( Y (14) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s ,s +2 − f ( s, = − f ( s, w j + m )+3 → j + m + s +1)+2 ⊗ w j → j +1) . (10) If r = 9 , then Ω ( Y (14) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b j, (2) s = κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1)+3 ⊗ e j . (11) If r = 10 , then Ω ( Y (14) t ) is described with (8 s × s ) -matrix with one nonzero elementthat is of the following form : b j + s − f ( s, , s +1 = − f ( s, κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1)+3 ⊗ w j +2 → j +3 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (14) t ) is a Ω r ( Y (14) t ) , whose left components twisted by σ ℓ . Proposition 31 (Translates for the case 15) . (I) Let r ∈ N , r < . r -translates of theelements Y (15) t are described by the following way. (1) If r = 0 , then Ω ( Y (15) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +3) ) e j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m )+1 → j + m )+2 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ e j + s )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .
08 MARIYA KACHALOVA If s j < s , then b ij = κ ℓ ( α j + m +3) ) e j + m )+3 ⊗ e j +3 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (2) If r = 1 , then Ω ( Y (15) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ e j , i = ( j + s ) s ;0 , otherwise.If s j < s , then b ij = κ w j + m )+2 → j + m )+3 ⊗ e j +1 , i = j ; κ e j + m )+3 ⊗ w j +1 → j +2 , i = j + 2 s ;0 , otherwise,where κ = − κ ℓ ( α j + m +3) ) .If s j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m +4) ) e j + m +1) ⊗ w j +2 → j +3 , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ e j +3 , i = j ;0 , otherwise.If s j < s , then b ij = 0 . (3) If r = 2 , then Ω ( Y (15) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ ( α j + m +2) ) e j + m ) − ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = − w j + m )+1 → j + m )+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +4) ) e j + m +1) ⊗ e j +3 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 109 (4) If r = 3 , then Ω ( Y (15) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ ( α j + m +3) ) w j + m )+2 → j + m )+3 ⊗ e j , i = j ; − κ ℓ ( α j + m +3) ) e j + m )+3 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +4) ) e j + m +1) ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ e j +3 , i = 6 s + ( j ) s ;0 , otherwise. (5) If r = 4 , then Ω ( Y (15) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ ( α j + m +3) ) e j + m ) ⊗ e j , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ e j + s )+2 , i = j + 2 s − sf ( j, s );0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +3) ) e j + m )+3 ⊗ e j +3 , i = j + 3 s ;0 , otherwise. (6) If r = 5 , then Ω ( Y (15) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m + s )+1 → j + m +1) ⊗ e j , i = j + s ; κ e j + m +1) ⊗ w j → j +1 , i = j + 2 s ; − κ w j + m )+3 → j + m +1) ⊗ w j → j +2 , i = j + 4 s ;0 , otherwise,
10 MARIYA KACHALOVA where κ = − κ ℓ ( α j + m +4) ) .If s j < s , then b ij = e j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + s + 1) s ; − w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + s + 1) s ; w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + s + 1) s ; κ e j + m +1)+3 ⊗ w j +3 → j +1)+2 , i = 4 s + ( j + 1) s ; − κ w j + m + s +1)+2 → j + m +1)+3 ⊗ e j +3 , i = j + s ;0 , otherwise,where κ = κ ℓ ( α j + m +4) ) . (7) If r = 6 , then Ω ( Y (15) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m )+1 ⊗ w j → j +1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w j + m ) → j + m )+1 ⊗ e j , i = j − s ; e j + m )+1 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +3) ) w j + m )+2 → j + m )+3 ⊗ e j +2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κ w j + m )+2 → j + m )+3 ⊗ e j +2 , i = j + s ; κ e j + m )+3 ⊗ w j +2 → j +3 , i = j + 2 s ;0 , otherwise,where κ = κ ℓ ( α j + m +3) ) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 111 If s j < s , then b ij = − f ( j, s − κ e j + m +1) ⊗ e j +3 , i = j + 2 s ; κ e j + m +1) ⊗ w j +3 → j +1) , i = ( j + 1) s , j < s − , otherwise,where κ = κ ℓ ( α j + m +4) ) . (8) If r = 7 , then Ω ( Y (15) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m + s )+2 → j + m )+3 ⊗ e j , i = j + s ; − κ e j + m )+3 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise,where κ = κ ℓ ( α j + m +3) ) .If s j < s , then b ij = e j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + s + 1) s ; − w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w j + m +1) → j + m +1)+1 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κ e j + m +2) ⊗ w j +3 → j + s +1)+2 , i = 4 s + ( j + s + 1) s ; κ w j + m + s +1)+1 → j + m +2) ⊗ e j +3 , i = j + s ; κ w j + m + s +1)+2 → j + m +2) ⊗ w j +3 → j +1) , i = s + ( j + 1) s , j = 6 s − − κ w j + m +1)+3 → j + m +2) ⊗ w j +3 → j +1)+1 , i = 2 s + ( j + 1) s , j = 6 s − , otherwise,where κ = κ ℓ +1 ( α j + m − ) . (9) If r = 8 , then Ω ( Y (15) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m )+2 ⊗ w j → j +1 , i = j + s ; w j + m )+1 → j + m )+2 ⊗ w j → j + s )+2 , i = j + 4 s ;0 , otherwise.
12 MARIYA KACHALOVA If s j < s , then b ij = − w j + m ) − → j + m )+2 ⊗ e j , i = j − s ; − e j + m )+2 ⊗ w j → j +1 , i = j + s ; − w j + m )+1 → j + m )+2 ⊗ w j → j + s )+2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +3) ) e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − or j = 4 s − , otherwise.If s j < s , then b ij = κ w j + m + s )+1 → j + m +1) ⊗ e j +2 , i = j − s ; − κ e j + m +1) ⊗ w j +2 → j +3 , i = j + s, j < s ;0 , otherwise,where κ = − κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = − w j + m +1) → j + m +1)+1 ⊗ e j +3 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (10) If r = 9 , then Ω ( Y (15) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ +1 ( α j + m − ) e j + m )+3 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = − κ ℓ +1 ( α j + m − ) e j + m +1) ⊗ w j → j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 113 If s j < s , then b ij = 0 .If s j < s , then b ij = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ e j +2 , i = j − s ;0 , otherwise.If s j < s , then b ij = κ w j + m +1)+2 → j + m +1)+3 ⊗ e j +3 , i = j − s ; − κ e j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s , j = 9 s − , otherwise,where κ = − κ ℓ +1 ( α j + m − ) . (11) If r = 10 , then Ω ( Y (15) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − w j + m ) → j + m )+1 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ w j + m + s )+1 → j + m +1) ⊗ e j +1 , i = j − s ; κ e j + m +1) ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − or j = 4 s − , otherwise,where κ = − f ( j, s ) κ ℓ ( α j + m − ) .If s j < s , then b ij = κ w j + m + s )+2 → j + m )+3 ⊗ e j +2 , i = j − s ; κ e j + m )+3 ⊗ w j +2 → j +3 , i = j + s, j < s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + s + 1) s ; − w j + m )+3 → j + m +1)+2 ⊗ e j +3 , i = j − s ;0 , otherwise.If s j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + s + 1) s ;0 , otherwise.
14 MARIYA KACHALOVA (II)
Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (15) t ) is a Ω r ( Y (15) t ) , whose left components twisted by σ ℓ . Proposition 32 (Translates for the case 16) . (I) Let r ∈ N , r < . r -translates of theelements Y (16) t are described by the following way. (1) If r = 0 , then Ω ( Y (16) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b , = w j + m ) → j + m )+2 ⊗ e j ; b s, s = w j + m )+1 → j + m )+3 ⊗ e j +1 . (2) If r = 1 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b j, = w j + m )+1 → j + m )+3 ⊗ e j ; b ( j +1) s , s − = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s − = e j + m + s +1)+2 ⊗ w j + s )+1 → j + s +1)+1 ; b ( j +1) s , s − = w j + m +1)+1 → j + m +2) ⊗ w j +3 → j +1) ; b j + s, s − = w j + m +1) → j + m +2) ⊗ e j +3 . (3) If r = 2 , then Ω ( Y (16) t ) is described with (7 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , s − = w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +1) ; b j, s − = w j + m + s )+2 → j + m +1) ⊗ e j + s )+1 ; b j, s − = w j + m + s )+2 → j + m +1) ⊗ e j + s )+1 ; b j, s − = w j + m )+1 → j + m +1)+1 ⊗ e j + s )+2 . (4) If r = 3 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b j + s,s − = w j + m + s )+2 → j + m +1) ⊗ e j ; b j +2 s,s − = w j + m )+3 → j + m +1) ⊗ w j → j +1 ; b j + s, s − = w j + m )+3 → j + m +1)+1 ⊗ e j + s )+1 . (5) If r = 4 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , ( s − s = w j + m )+3 → j + m +1) ⊗ w j → j +1) ; b s +( j +1) s , ( s − s = e j + m +1) ⊗ w j → j +1) ; b ( j +1) s , s − (2) s = w j + m )+3 → j + m +1)+1 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s − (2) s = w j + m +1) → j + m +1)+1 ⊗ w j + s )+1 → j +1) ; b j +2 s + f ( s, , s − (2) s = w j + m )+2 → j + m +1)+2 ⊗ e j + s )+2 . (6) If r = 5 , then Ω ( Y (16) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s , s − = e j + m + s +1)+1 ⊗ w j → j +1) ; b j, s − (2) s = w j + m +1) → j + m + s +1)+2 ⊗ e j +1 . (7) If r = 6 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b j + s +2 f ( s, , ( s − s = w j + m + f ( s, → j + m )+3 ⊗ w j → j + f ( s, ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 115 b ( j +1) s , s − f ( s, = w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s − = w j + m + s +1)+1 → j + m +2) ⊗ w j +3 → j + s +1)+1 , s > . (8) If r = 7 , then Ω ( Y (16) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b s +( j +1) s − f ( s, , ( s − s = e j + m + s +1)+2 ⊗ w j → j +1) ; b ( j +1) s + f ( s, , s +( s − s = w j + m +1)+2 → j + m +1)+3 ⊗ w j +1 → j +1) ; b s +( j +1) s + f ( s, , s +( s − s = w j + m +1)+3 → j + m + s +2)+1 ⊗ w j +3 → j + s +1)+1 ; b s +( j +1) s + f ( s, , s +( s − s = w j + m +2) → j + m + s +2)+1 ⊗ w j +3 → j + s +1)+2 ; b s +( j +1) s − f ( s, , s +( s − s = w j + m +2) → j + m + s +2)+1 ⊗ w j +3 → j +1)+2 . (9) If r = 8 , then Ω ( Y (16) t ) is described with (9 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , ( s − s = e j + m )+3 ⊗ w j → j +1) ; b ( j +1) s ,s +( s − s = w j + m )+3 → j + m +1) ⊗ w j → j +1) ; b ( j +1) s , s +( s − s = w j + m )+3 → j + m +1)+1 ⊗ w j +1 → j +1) ; b s +( j +1) s + f ( s, , s +( s − s = e j + m + s +1)+2 ⊗ w j + s )+2 → j + s +1)+1 ; b s +( j +1) s − f ( s, , s +( s − s = e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+2 ; b ( j +1) s , s +( s − s = w j + m )+3 → j + m +1)+3 ⊗ w j +3 → j +1) ; b s +( j +1) s − f ( s, , s +( s − s = w j + m + s +1)+2 → j + m +1)+3 ⊗ w j +3 → j + s +1)+1 ; b s +( j +1) s + f ( s, , s +( s − s = w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1)+2 . (10) If r = 9 , then Ω ( Y (16) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , s − f ( s, = w j + m +1)+3 → j + m +2) ⊗ w j +1 → j +1) ; b ( j +1) s , s − f ( s, = e j + m +1)+3 ⊗ w j +2 → j +1) ; b ( j +1) s , s − f ( s, = e j + m +1)+3 ⊗ w j +2 → j +1) ; b ( j +1) s , s − f ( s, = w j + m +1)+3 → j + m + s +2)+2 ⊗ w j +3 → j +1) . (11) If r = 10 , then Ω ( Y (16) t ) is described with (9 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , ( s − s = e j + m +1) ⊗ w j → j +1) ; b ( j +1) s ,s +( s − s = w j + m +1) → j + m +1)+1 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s + f ( s, ,s +( s − s = e j + m +1)+1 ⊗ w j + s )+1 → j + s +1)+1 ; b j − s, s +( s − s = w j + m )+2 → j + m +1)+2 ⊗ e j + s )+2 ; b j − s, s +( s − s = w j + m )+3 → j + m +1)+2 ⊗ w j + s )+2 → j +3 ; b j − s, s +( s − s = w j + m )+3 → j + m +1)+3 ⊗ e j +3 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (16) t ) is a Ω r ( Y (16) t ) , whose left components twisted by σ ℓ .
16 MARIYA KACHALOVA
Proposition 33 (Translates for the case 17) . (I) Let r ∈ N , r < . r -translates of theelements Y (17) t are described by the following way. (1) If r = 0 , then Ω ( Y (17) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b , = w j + m ) → j + m + s )+2 ⊗ e j ; b ,s = w j + m ) → j + m + s )+2 ⊗ e j . (2) If r = 1 , then Ω ( Y (17) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b s +( j +1) s − f ( s, ,s +(1) s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b ( j +1) s + f ( s, , s +(1) s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b j + s, s +(1) s = − κ ℓ ( α j + m +5) ) w j + m +1) → j + m +2) ⊗ e j +3 . (3) If r = 2 , then Ω ( Y (17) t ) is described with (7 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , (1) s = f ( s, κ ℓ ( α j + m +3) ) e j + m )+3 ⊗ w j → j +1) ; b ( j +1) s ,s +(1) s = − f ( s, κ ℓ ( α j + m +4) ) w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +1) ; b j,s +(1) s = − κ ℓ ( α j + m +4) ) w j + m + s )+2 → j + m +1) ⊗ e j + s )+1 ; b ( j +1) s , s +(1) s = f ( s, κ ℓ ( α j + m +4) ) w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +1) ; b j, s +(1) s = − κ ℓ ( α j + m +4) ) w j + m + s )+2 → j + m +1) ⊗ e j + s )+1 . (4) If r = 3 , then Ω ( Y (17) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b j + s, (1) s = κ ℓ ( α j + m +4) ) w j + m + s )+2 → j + m +1) ⊗ e j ; b j +2 s, (1) s = − κ ℓ ( α j + m +4) ) w j + m )+3 → j + m +1) ⊗ w j → j +1 ; b j + s,s +(1) s = − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+1 ; b j + s, s +(1) s = − w j + m )+3 → j + m + s +1)+1 ⊗ e j + s )+1 . (5) If r = 4 , then Ω ( Y (17) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , = − f ( s, κ ℓ ( α j + m +4) ) w j + m )+3 → j + m +1) ⊗ w j → j +1) ; b s +( j +1) s , = f ( s, κ ℓ ( α j + m +4) ) e j + m +1) ⊗ w j → j +1) ; b ( j +1) s , s − f ( s, = w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) ; b ( j +1) s , s − f ( s, = − w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+2 → j +1) ; b ( j +1) s , s = − f ( s, κ ℓ ( α j + m +4) ) w j + m )+3 → j + m +1)+3 ⊗ w j +3 → j +1) ; b s +( j +1) s , s = f ( s, κ ℓ ( α j + m +4) ) w j + m +1+ f ( s, → j + m +1)+3 ⊗ w j +3 → j + s +1+ f ( s, ; b s +( j +1) s , s = − f ( s, κ ℓ ( α j + m +4) ) w j + m +1+ f ( s, → j + m +1)+3 ⊗ w j +3 → j + s +1+ f ( s, . (6) If r = 5 , then Ω ( Y (17) t ) is described with (7 s × s ) -matrix with the following nonzeroelements : b ( j +1) s + f ( s, , = − e j + m +1)+1 ⊗ w j → j +1) ; b s +( j +1) s − f ( s, ,s = − e j + m +1)+1 ⊗ w j → j +1) ; b ( j +1) s , s + f ( s, = f ( s, κ ℓ ( α j + m +4) ) w j + m +1)+1 → j + m +1)+3 ⊗ w j +2 → j +1) ; b s +( j +1) s , s + f ( s, = f ( s, κ ℓ ( α j + m +4) ) w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +2 → j +1) ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 117 b s +( j +1) s , s + f ( s, = − f ( s, κ ℓ ( α j + m +4) ) e j + m +1)+3 ⊗ w j +2 → j +1)+2 ; b s +( j +1) s , s = κ ℓ ( α j + m +5) ) e j + m +2) ⊗ w j +3 → j +1+ f ( s, . (7) If r = 6 , then Ω ( Y (17) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j +1) s ,s + f ( s, = w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) ; b ( j +1) s , s + f ( s, = w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +1) ; b s +( j +1) s , s = f ( s, κ ℓ +1 ( α j + m − ) w j + m + s +1+ f ( s, → j + m +2) ⊗ w j +3 → j + s +1+ f ( s, . (8) If r = 7 , then Ω ( Y (17) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b ( j +1) s + f ( s, , = − e j + m +1)+2 ⊗ w j → j +1) ; b s +( j +1) s − f ( s, ,s = e j + m +1)+2 ⊗ w j → j +1) ; b ( j +1) s , s + f ( s, = κ ℓ +1 ( α j + m − ) w j + m +1)+2 → j + m +2) ⊗ w j +2 → j +1) ; b s +( j +1) s , s + f ( s, = − κ ℓ +1 ( α j + m − ) e j + m +2) ⊗ w j +2 → j +1)+2 ; b s +( j +1) s , s + f ( s, = w j + m +1)+3 → j + m +2)+1 ⊗ w j +3 → j +1)+1 ; b s +( j +1) s , s + f ( s, = w j + m +2) → j + m +2)+1 ⊗ w j +3 → j +1)+2 ; b s +( j +1) s , s + f ( s, = w j + m +2) → j + m +2)+1 ⊗ w j +3 → j + s +1)+2 . (9) If r = 8 and s = 1 , then Ω ( Y (17) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b j, = − κ ℓ +1 ( α j + m − ) e j + m )+3 ⊗ e j ; b j − s,s = κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m ) ⊗ e j ; b j − s, s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +1)+2 → j +2 ; b j − s, s = w j + m )+3 → j + m )+1 ⊗ w j +1)+2 → j ; b j − s, s = − w j + m )+1 → j + m )+2 ⊗ w j +2 → j +1)+2 ; b j − s, s = − κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1)+3 ⊗ w j +3 → j ; b j − s, s = − κ ℓ +1 ( α j + m − ) w j + m +1)+2 → j + m )+3 ⊗ w j +3 → j +1)+1 ; b j − s, s = − κ ℓ +1 ( α j + m − ) w j + m +1)+1 → j + m )+3 ⊗ w j +3 → j +2 . (10) If r = 8 and s > , then Ω ( Y (17) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b ( j +1) s , = κ ℓ +1 ( α j + m − ) e j + m )+3 ⊗ w j → j +1) ; b ( j +1) s , s = w j + m )+3 → j + m +1)+1 ⊗ w j +2 → j +1) ; b s +( j +1) s , s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j + s +1)+2 ; b s +( j +1) s , s = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +2 → j +1)+2 ; b ( j +1) s , s = κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1)+3 ⊗ w j +3 → j +1) ; b s +( j +1) s , s = κ ℓ +1 ( α j + m − ) w j + m + s +1)+2 → j + m +1)+3 ⊗ w j +3 → j + s +1)+1 ; b s +( j +1) s , s = κ ℓ +1 ( α j + m − ) w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1)+2 . (11) If r = 9 and s = 1 , then Ω ( Y (17) t ) is described with (8 s × s ) -matrix with the followingtwo nonzero elements : b j − s, s = − κ ℓ +1 ( α j + m − ) e j + m )+3 ⊗ w j +2 → j ; b j − s, s = w j + m )+3 → j + m )+2 ⊗ w j +3 → j .
18 MARIYA KACHALOVA (12) If r = 9 and s > , then Ω ( Y (17) t ) is described with (8 s × s ) -matrix with the followingtwo nonzero elements : b ( j +1) s , s = − κ ℓ +1 ( α j + m − ) e j + m +1)+3 ⊗ w j +2 → j +1) ; b ( j +1) s , s = − w j + m +1)+3 → j + m +2)+2 ⊗ w j +3 → j +1) . (13) If r = 10 and s = 1 , then Ω ( Y (17) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b j, = − κ ℓ +1 ( α j + m − ) e j + m ) ⊗ e j ; b j + s, s = w j + m +1)+2 → j + m )+2 ⊗ w j +1 → j +2 ; b j +3 s, s = w j + m )+3 → j + m )+2 ⊗ w j +1 → j +3 ; b j, s = w j + m )+2 → j + m +1)+2 ⊗ w j +1)+1 → j +1)+2 ; b j + s, s = − w j + m )+3 → j + m +1)+2 ⊗ w j +1)+1 → j +3 ; b j − s, s = − κ ℓ +1 ( α j + m − ) e j + m )+3 ⊗ e j +3 . (14) If r = 10 and s > , then Ω ( Y (17) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b ( j +1) s , = κ ℓ +1 ( α j + m − ) e j + m +1) ⊗ w j → j +1) ; b j + s, s = w j + m + s )+2 → j + m + s +1)+2 ⊗ w j +1 → j +2 ; b j +3 s, s = w j + m )+3 → j + m + s +1)+2 ⊗ w j +1 → j +3 ; b j, s = w j + m )+2 → j + m +1)+2 ⊗ w j + s )+1 → j + s )+2 ; b j + s, s = − w j + m )+3 → j + m +1)+2 ⊗ w j + s )+1 → j +3 ; b j − s, s = − κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1)+3 ⊗ e j +3 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (17) t ) is a Ω r ( Y (17) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 34 (Translates for the case 18) . (I) Let r ∈ N , r < . r -translates of theelements Y (18) t are described by the following way. (1) If r = 0 , then Ω ( Y (18) t ) is described with (6 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m +4+1) ) w j + m )+3 → j + m +1) ⊗ e j +3 , i = j ;0 , otherwise. (2) If r = 1 , then Ω ( Y (18) t ) is described with (7 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 119 If j < s , then b ij = κ w j + m )+1 → j + m )+3 ⊗ e j , i = j ; − κ w j + m + s )+1 → j + m )+3 ⊗ e j , i = j + s ;0 , otherwise,where κ = κ ℓ ( α j +4+ m ) ) .If s j < s , then b ij = κ w j + m + s )+2 → j + m +1) ⊗ e j + s )+1 , i = j + s ; κ w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise,where κ = κ ℓ ( α j + m +4+1) ) .If s j < s , then b ij = − w j + m )+3 → j + m +1)+1 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j +3 → j +1) , i = ( j + s + 1) s ; − w j + m +1) → j + m + s +1)+2 ⊗ e j +3 , i = 6 s + ( j ) s ;0 , otherwise. (3) If r = 2 , then Ω ( Y (18) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +4) ) w j + m ) − → j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w j + m )+1 → j + m )+2 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = 0 . (4) If r = 3 , then Ω ( Y (18) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m )+2 → j + m +1) ⊗ e j , i = j ; κ w j + m )+3 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise,where κ = κ ℓ ( α j + m +4+1) ) .
20 MARIYA KACHALOVA If s j < s , then b ij = − w j + m )+3 → j + m +1)+1 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − f ( j, s − e j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = f ( j, s − e j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; κ e j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ;0 , otherwise,where κ = − κ ℓ ( α j + m +4+1) ) f ( j, s − . (5) If r = 4 , then Ω ( Y (18) t ) is described with (7 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m ) − → j + m + s )+1 ⊗ e j , i = j − s ;0 , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ e j +1 , i = j ; − e j + m + s )+2 ⊗ w j +1 → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m + s )+1 → j + m + s )+2 ⊗ e j +1 , i = j ; − e j + m + s )+2 ⊗ w j +1 → j +2 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j +4+ m ) ) f ( j, s − e j + m )+3 ⊗ w j +2 → j +1) ,i = ( j + 1) s , j < s − or j = 6 s − , otherwise.If s j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 121 (6) If r = 5 , then Ω ( Y (18) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m )+1 → j + m )+3 ⊗ e j , i = j ; κ w j + m + s )+1 → j + m )+3 ⊗ e j , i = j + s ; − κ e j + m )+3 ⊗ w j → j +2 , i = j + 4 s ; − κ e j + m )+3 ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise,where κ = − κ ℓ ( α j +4+ m ) ) .If s j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; − w j + m +1) → j + m +1)+2 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ +1 ( α j + m ) ) f ( j, s − e j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ;0 , otherwise. (7) If r = 6 , then Ω ( Y (18) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) → j + m + s )+2 ⊗ e j , i = j ; − w j + m + s )+1 → j + m + s )+2 ⊗ w j → j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w j + m ) → j + m + s )+2 ⊗ e j , i = j − s ; w j + m + s )+1 → j + m + s )+2 ⊗ w j → j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κ ℓ ( α j + m +4) ) f (( j ) s , s − w j + m )+1 → j + m )+3 ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +4) ) f (( j ) s , s − w j + m )+1 → j + m )+3 ⊗ e j +1 , i = j ;0 , otherwise.
22 MARIYA KACHALOVA If s j < s , then b ij = − κ ℓ +1 ( α j + m − ) e j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s , j < s − or j = 6 s − , otherwise.If s j < s − , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = − e j + m + s +1)+1 ⊗ w j +3 → j + s +1)+1 , i = 3 s + ( j + 1) s − sf ( j, s − , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − e j + m + s +1)+1 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + 1) s + 2 sf ( j, s − , otherwise. (8) If r = 7 , then Ω ( Y (18) t ) is described with (9 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = − κ w j + m )+2 → j + m +1) ⊗ e j , i = j ; κ w j + m )+3 → j + m +1) ⊗ w j → j + s )+1 , i = j + s ; κ e j + m +1) ⊗ w j → j + s )+2 , i = j + 3 s ; κ e j + m +1) ⊗ w j → j +2 , i = j + 4 s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = w j + m )+3 → j + m +1)+1 ⊗ e j +1 , i = j ; w j + m +1) → j + m +1)+1 ⊗ w j +1 → j +2 , i = j + 2 s ; − e j + m +1)+1 ⊗ w j +1 → j +3 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 123 (9) If r = 8 , then Ω ( Y (18) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) − → j + m + s )+1 ⊗ e j , i = j ; e j + m + s )+1 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = e j + m + s )+1 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1) ⊗ w j +1 → j +1) ,i = ( j + 1) s , j < s − or j = 4 s − κ ℓ +1 ( α j + m − ) w j + m )+2 → j + m +1) ⊗ e j +1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − f ( j, s ) e j + m + s +1)+2 ⊗ w j +3 → j + s +1)+1 , i = s + ( j + s + 1) s ;0 , otherwise. (10) If r = 9 , then Ω ( Y (18) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ +1 ( α j + m − ) w j + m )+3 → j + m +1) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m +1) → j + m +1)+2 ⊗ e j +1 , i = j − s ; − e j + m +1)+2 ⊗ w j +1 → j +3 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m +1) → j + m + s +1)+2 ⊗ e j + s )+1 , i = j − s ; − e j + m + s +1)+2 ⊗ w j + s )+1 → j +3 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .
24 MARIYA KACHALOVA If s j < s , then b ij = − κ ℓ +1 ( α j + m − ) f ( j, s − e j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (18) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) → j + m + s )+2 ⊗ e j , i = j ; − w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +1 , i = j + s ; e j + m + s )+2 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w j + m ) → j + m + s )+2 ⊗ e j , i = j − s ; − w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +1 , i = j + s ; e j + m + s )+2 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − κ w j + m + s )+1 → j + m )+3 ⊗ e j +1 , i = j − s ; κ w j + m + s )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + s ; κ e j + m )+3 ⊗ w j +1 → j +3 , i = j + 3 s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = − κ w j + m + s )+1 → j + m )+3 ⊗ e j +1 , i = j − s ; κ w j + m + s )+2 → j + m )+3 ⊗ w j +1 → j +2 , i = j + s ; − κ e j + m )+3 ⊗ w j +1 → j +3 , i = j + 2 s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = 0 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (18) t ) is a Ω r ( Y (18) t ) , whose left components twisted by σ ℓ . Proposition 35 (Translates for the case 19) . (I) Let r ∈ N , r < . r -translates of theelements Y (19) t are described by the following way. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 125 (1) If r = 0 , then Ω ( Y (19) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b s,s = κ ℓ ( α j +5) ) w j + m + s )+1 → j + m +1) ⊗ e j + s )+1 ; b s, s = κ ℓ ( α j +5) ) w j + m + s )+1 → j + m +1) ⊗ e j + s )+1 . (2) If r = 1 , then Ω ( Y (19) t ) is described with (6 s × s ) -matrix with the following two nonzeroelements : b j, (3) s = − κ ℓ ( α j + m +5) ) w j + m )+1 → j + m +1) ⊗ e j ; b j + s, (3) s = κ ℓ ( α j + m +5) ) w j + m + s )+1 → j + m +1) ⊗ e j . (3) If r = 2 , then Ω ( Y (19) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , (2) s = − f ( s, κ ℓ ( α j + m +5) ) w j + m )+3 → j + m +1) ⊗ w j → j +1) ; b ( j +1) s ,s +(2) s = f ( s, w j + m )+3 → j + m +1)+1 ⊗ w j + s )+1 → j +1) ; b ( j +1) s , s +(2) s = − f ( s, w j + m )+3 → j + m +1)+1 ⊗ w j + s )+1 → j +1) . (4) If r = 3 , then Ω ( Y (19) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b s +( j +1) s − f ( s, , s +(2) s = − f ( s, e j + m + s +1)+2 ⊗ w j +1 → j +1) ; b ( j +1) s + f ( s, , s +(2) s = f ( s, e j + m + s +1)+2 ⊗ w j +1 → j +1) . (5) If r = 4 , then Ω ( Y (19) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b ( j +1) s , (2) s = − κ ℓ ( α j + m +4) ) e j + m )+3 ⊗ w j → j +1) ; b ( j +1) s ,s +(2) s + f ( s, = w j + m )+3 → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s ,s +(2) s + f ( s, = − w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s , s +(2) s − f ( s, = w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j +1) . (6) If r = 5 , then Ω ( Y (19) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b s +( j +1) s − f ( s, , (2) s = w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j → j +1) ; b ( j +1) s + f ( s, ,s +(2) s = − w j + m + s +1)+1 → j + m + s +1)+2 ⊗ w j → j +1) ; b ( j +1) s + f ( s, , s +(2) s = − κ ℓ +1 ( α j + m − ) w j + m +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) ; b s +( j +1) s − f ( s, , s +(2) s = − κ ℓ +1 ( α j + m − ) w j + m +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) . (7) If r = 6 , then Ω ( Y (19) t ) is described with (9 s × s ) -matrix with the following nonzeroelements : b ( j +1) s ,s +(2) s = − κ ℓ +1 ( α j + m − ) e j + m +1) ⊗ w j → j +1) ; b j − s, s +(2) s = w j + m )+1 → j + m +1)+1 ⊗ e j +1 ; b j, s +(2) s = w j + m )+1 → j + m +1)+1 ⊗ e j +1 . (8) If r = 7 , then Ω ( Y (19) t ) is described with (8 s × s ) -matrix with the following nonzeroelements : b j + s, (2) s = w j + m + s )+2 → j + m + s +1)+1 ⊗ e j ; b j +2 s, (2) s = − w j + m )+3 → j + m + s +1)+1 ⊗ w j → j +1 ; b j +4 s, (2) s = − w j + m +1) → j + m + s +1)+1 ⊗ w j → j +2 ; b j +5 s, (2) s = − w j + m +1) → j + m + s +1)+1 ⊗ w j → j + s )+2 ;
26 MARIYA KACHALOVA b j + s,s +(2) s = − w j + m )+3 → j + m + s +1)+1 ⊗ w j → j + s )+1 ; b j +3 s,s +(2) s = − w j + m +1) → j + m + s +1)+1 ⊗ w j → j + s )+2 ; b j +4 s,s +(2) s = − w j + m +1) → j + m + s +1)+1 ⊗ w j → j +2 . (9) If r = 8 and s = 2 , then Ω ( Y (19) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b j − s, s +1 = w j + m )+2 → j + m +1)+2 ⊗ e j +1 ; b ( j +1) s , s +1 = w j + m )+3 → j + m + s +1)+1 ⊗ w j +1 → j +1) ; b s +( j +1) s , s +1 = e j + m + s +1)+2 ⊗ w j +2)+1 → j + s +1)+1 . (10) If r = 8 and s = 2 , then Ω ( Y (19) t ) is described with (9 s × s ) -matrix with the followingnonzero elements : b ( j +1) s ,s +1+ f ( s, = w j + m )+3 → j + m +1+ f ( s, ⊗ w j + s + f ( s, → j +1) ; b s +( j +1) s , s +1+ f ( s, = e j + m +1+ f ( s, ⊗ w j + f ( s, → j +1+ f ( s, ; b j − s + f ( s, , s +1 − f ( s, = w j + m + s + f ( s, → j + m + s +1+ f ( s, ⊗ e j + s + f ( s, . (11) If r = 9 and s = 2 , then Ω ( Y (19) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j,s +1 = − w j + m +1) → j + m + s +1)+2 ⊗ w j → j +2)+1 ; b j + s,s +1 = w j + m +1) → j + m + s +1)+2 ⊗ w j → j +1 ; b ( j +1) s , s +1 = e j + m +1)+3 ⊗ w j +1 → j +1) . (12) If r = 9 and s = 2 , then Ω ( Y (19) t ) is described with (8 s × s ) -matrix with the followingnonzero elements : b j − f ( s, , = f ( s, w j + m )+3 → j + m + s +1)+2 ⊗ e j ; b j − s, s +1 = κ ℓ +1 ( α j + m − ) w j + m +1) → j + m +1)+3 ⊗ e j +1 ; b j +3 s, s +1 = − κ ℓ +1 ( α j + m − ) w j + m +1)+2 → j + m +1)+3 ⊗ w j +1 → j +3 . (13) If r = 10 and s = 1 , then Ω ( Y (19) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j, = κ ℓ +1 ( α j + m − ) w j + m ) → j + m )+3 ⊗ e j ; b j + s, = − κ ℓ +1 ( α j + m − ) w j + m +1)+1 → j + m )+3 ⊗ w j → j +1 ; b j +3 s, = κ ℓ +1 ( α j + m − ) w j + m +1)+2 → j + m )+3 ⊗ w j → j +2 ; b j − s,s = − w j + m ) → j + m +1)+2 ⊗ w j +1)+1 → j ; b j + s,s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +1)+1 → j +1 . (14) If r = 10 and s = 2 , then Ω ( Y (19) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j + s, = − w j + m +2)+1 → j + m )+3 ⊗ w j → j +1 ; b j +2 s, = w j + m )+1 → j + m )+3 ⊗ w j → j +2)+1 ; b j +3 s, = w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 ; b j +4 s, = − w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 ; b j +5 s, = e j + m )+3 ⊗ w j → j +3 ; b s +( j +1) s , s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j +2)+1 → j + s +1)+1 ; b j +2 s, s +1 = w j + m )+2 → j + m +1)+2 ⊗ w j +2)+1 → j + s )+2 ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 127 b j +3 s, s +1 = − w j + m )+3 → j + m +1)+2 ⊗ w j +2)+1 → j +3 ; b ( j +1) s , s = − w j + m +1) → j + m ) ⊗ w j +3 → j +1) . (15) If r = 10 and s > , then Ω ( Y (19) t ) is described with (6 s × s ) -matrix with the followingnonzero elements : b j, = κ ℓ +1 ( α j + m − ) w j + m ) → j + m )+3 ⊗ e j ; b j +2 s, = κ ℓ +1 ( α j + m − ) w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 ; b j +4 s, = − κ ℓ +1 ( α j + m − ) w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 ; b ( j +1) s ,s = w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j +1) ; b s +( j +1) s ,s = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+1 → j + s +1)+1 ; b ( j +1) s , s = − κ ℓ +1 ( α j + m − ) w j + m +1) → j + m +2) ⊗ w j +3 → j +1) ; b s +( j +1) s , s = − κ ℓ +1 ( α j + m − ) w j + m + s +1)+1 → j + m +2) ⊗ w j +3 → j +1)+1 ; b s +( j +1) s , s = κ ℓ +1 ( α j + m − ) w j + m + s +1)+2 → j + m +2) ⊗ w j +3 → j +1)+2 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (19) t ) is a Ω r ( Y (19) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 36 (Translates for the case 20) . (I) Let r ∈ N , r < . r -translates of theelements Y (20) t are described by the following way. (1) If r = 0 , then Ω ( Y (20) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +4) ) w j + m ) → j + m )+3 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +4+1) ) w j + m + s )+1 → j + m +1) ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m + s )+2 → j + m + s +1)+1 ⊗ e j + s )+2 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m )+3 → j + m +1)+2 ⊗ e j +3 , i = j − s ;0 , otherwise. (2) If r = 1 , then Ω ( Y (20) t ) is described with (6 s × s ) -matrix with the following elements b ij :
28 MARIYA KACHALOVA If j < s , then b ij = − κ ℓ ( α j + m +5) ) w j + m )+1 → j + m +1) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = − w j + m + s )+2 → j + m + s +1)+1 ⊗ e j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +5) ) w j + m +1) → j + m +1)+3 ⊗ e j +3 , i = j + s ;0 , otherwise. (3) If r = 2 , then Ω ( Y (20) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m )+2 → j + m +1) ⊗ w j → j +1 , i = j + s ; κ w j + m )+3 → j + m +1) ⊗ w j → j +1) , i = ( j + 1) s , j = s − , otherwise,where κ = κ ℓ ( α j + m +5) ) .If s j < s − , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 129 If s − j < s , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − κ ℓ ( α j + m +5) ) w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − ,i = s + ( j + 1) s ;0 , otherwise. (4) If r = 3 , then Ω ( Y (20) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+3 → j + m +1)+1 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ ( α j + m +5) ) f ( j, s ) w j + m +1)+2 → j + m +1)+3 ⊗ w j +2 → j +1) ,i = ( j + 1) s , j < s − or j = 6 s − , otherwise.If s j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (20) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m + s )+1 → j + m )+3 ⊗ w j → j +1 , i = j + 2 s ; − κ w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 , i = j + 5 s ; κ e j + m )+3 ⊗ w j → j +1) , i = ( j + 1) s , j < s − , otherwise,where κ = κ ℓ ( α j + m +4) ) .If s j < s , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s , j < s − or j = 3 s − , otherwise.If s j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (20) t ) is described with (8 s × s ) -matrix with the following elements b ij : If s j < s , then b ij = w j + m +1)+1 → j + m +1)+2 ⊗ w j → j +1) , i = ( j + 1) s ; w j + m +1) → j + m +1)+2 ⊗ w j → j + s )+1 , i = j + s ;0 , otherwise.
30 MARIYA KACHALOVA If s j < s , then b ij = κ w j + m +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) ,i = ( j + 1) s , j < s − or j = 4 s − κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +1 → j +1) ,i = s + ( j + 1) s , j < s − or j = 4 s − , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = κ ℓ +1 ( α j + m ) ) e j + m +2) ⊗ w j +2 → j +1)+1 , i = 2 s + ( j + 1) s , j < s − or j = 6 s − , otherwise.If s j < s , then b ij = 0 . (7) If r = 6 , then Ω ( Y (20) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ ℓ +1 ( α j + m − ) w j + m )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ;0 , otherwise.If s j < s − , then b ij = κ ℓ +1 ( α j + m − ) e j + m +1) ⊗ w j → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − or j = 4 s − , otherwise.If s j < s − , then b ij = e j + m +1)+1 ⊗ w j +2 → j +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = e j + m + s +1)+1 ⊗ w j + s )+2 → j + s +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (20) t ) is described with (8 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 131 If j < s , then b ij = − w j + m )+3 → j + m +1)+1 ⊗ w j → j +1 , i = j + 2 s ; − w j + m +1) → j + m +1)+1 ⊗ w j → j +2 , i = j + 4 s ; − w j + m +1) → j + m +1)+1 ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ +1 ( α j + m ) ) w j + m +1)+2 → j + m +2) ⊗ w j +1 → j +1) ,i = ( j + 1) s , j < s − or j = 4 s − , otherwise.If s j < s , then b ij = 0 . (9) If r = 8 , then Ω ( Y (20) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κ w j + m )+2 → j + m +1) ⊗ w j → j +1 , i = j + s ; − κ w j + m )+1 → j + m +1) ⊗ w j → j + s )+2 , i = j + 4 s ; κ w j + m )+3 → j + m +1) ⊗ w j → j +1) , i = ( j + 1) s , j = s − , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s − , then b ij = w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m )+3 → j + m +1)+1 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ w j + m )+3 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 3 s + ( j + 1) s ;0 , otherwise,where κ = − κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = 0 .
32 MARIYA KACHALOVA (10) If r = 9 , then Ω ( Y (20) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m )+3 → j + m +1)+2 ⊗ e j , i = j ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s − , then b ij = − κ ℓ +1 ( α j + m − ) e j + m +1)+3 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ +1 ( α j + m ) ) w j + m +1)+3 → j + m +2) ⊗ w j +2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = − w j + m +1)+3 → j + m +2)+1 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = − w j + m +2) → j + m +2)+1 ⊗ w j +3 → j +1)+1 , i = 2 s + ( j + 1) s ; − e j + m +2)+1 ⊗ w j +3 → j + s +1)+2 , i = 3 s + ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (20) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m ) → j + m )+3 ⊗ e j , i = j ; − κ w j + m + s )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ; κ w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 2 s ; κ w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 , i = j + 3 s ; − κ w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 , i = j + 4 s ; κ e j + m )+3 ⊗ w j → j +3 , i = j + 5 s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s − , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 133 If s − j < s , then b ij = − w j + m +1) → j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1) → j + m +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; − e j + m +1)+1 ⊗ w j + s )+2 → j + s +1)+1 , i = s + ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = κ w j + m +1)+1 → j + m +2) ⊗ w j +3 → j + s +1)+1 , i = 2 s + ( j + 1) s ; κ w j + m + s +1)+2 → j + m +2) ⊗ w j +3 → j +1)+2 , i = 3 s + ( j + 1) s ; − κ w j + m +1)+2 → j + m +2) ⊗ w j +3 → j + s +1)+2 , i = 4 s + ( j + 1) s ; κ w j + m +1)+3 → j + m +2) ⊗ w j +3 → j +1)+3 , i = 5 s + ( j + 1) s ;0 , otherwise,where κ = − f ( s, κ ℓ +1 ( α j + m +1) ) . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (20) t ) is a Ω r ( Y (20) t ) , whose left components twisted by σ ℓ . Proposition 37 (Translates for the case 21) . (I) Let r ∈ N , r < . Denote by κ = κ ℓ ( γ ) κ ℓ ( γ ) κ ℓ ( γ ) − κ ℓ ( α ) . Then r -translates of the elements Y (21) t are described by the fol-lowing way. (1) If r = 0 , then Ω ( Y (21) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b , = − κ ℓ ( α ) w j + m ) → j + m +1) ⊗ e j . (2) If r = 1 , then Ω ( Y (21) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , (1) s = f ( s, κ w j + m +1+ f ( s, → j + m +2) ⊗ w j → j +1) . (3) If r = 2 , then Ω ( Y (21) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , = f ( s, κ ℓ +1 ( α j + m − ) κ w j + m )+3 → j + m +1)+1 ⊗ w j → j +1) . (4) If r = 3 , then Ω ( Y (21) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , (1) s = − κ w j + m +1+ f ( s, → j + m +1)+3 ⊗ w j → j +1) .
34 MARIYA KACHALOVA (5) If r = 4 , then Ω ( Y (21) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , = − κ ℓ +1 ( α j + m − ) κ w j + m )+3 → j + m +1)+2 ⊗ w j → j +1) . (6) If r = 5 , then Ω ( Y (21) t ) is described with (9 s × s ) -matrix with the following two nonzeroelements : b ( j +1) s + f ( s, , (1) s = f ( s, κ w j + m +1)+1 → j + m +1)+3 ⊗ w j → j +1) ; b s +( j +1) s − f ( s, , (1) s = f ( s, κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j → j +1) . (7) If r = 6 , then Ω ( Y (21) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , (1) s = f ( s, κ ℓ +1 ( α j + m − ) κ w j + m +1) → j + m +1)+1 ⊗ w j → j +1) . (8) If r = 7 , then Ω ( Y (21) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , (1) s = f ( s, κ ℓ +1 ( γ j + m − ) κ w j + m +1+ f ( s, → j + m +2) ⊗ w j → j +1) . (9) If r = 8 , then Ω ( Y (21) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b ( j +1) s , = f ( s, κ ℓ +1 ( α j + m − ) κ w j + m )+3 → j + m +1)+2 ⊗ w j → j +1) . (10) If r = 9 , then Ω ( Y (21) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, (1) s = − f ( s, κ ℓ +1 ( γ j + m − ) κ w j + m )+3 → j + m +1)+3 ⊗ e j . (11) If r = 10 , then Ω ( Y (21) t ) is described with (7 s × s ) -matrix with one nonzero elementthat is of the following form : b ( j +1) s , = − κ ℓ +1 ( γ j + m − ) κ w j + m +1) → j + m +1)+3 ⊗ w j → j +1) . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (21) t ) is a Ω r ( Y (21) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 38 (Translates for the case 22) . (I) Let r ∈ N , r < . r -translates of theelements Y (22) t are described by the following way. (1) If r = 0 , then Ω ( Y (22) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ ( α j + m +5) ) e j + m ) ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κ ℓ ( α j + m +5) ) e j + m )+3 ⊗ e j +3 , i = j ;0 , otherwise. (2) If r = 1 , then Ω ( Y (22) t ) is described with (6 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 135 If j < s , then b ij = κ w j + m )+1 → j + m +1) ⊗ e j , i = j ; κ w j + m )+2 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ; κ w j + m )+3 → j + m +1) ⊗ w j → j +2 , i = j + 4 s ; − κ e j + m +1) ⊗ w j → j +3 , i = j + 6 s ;0 , otherwise,where κ = − κ ℓ ( α j + m +6) ) .If s j < s , then b ij = − e j + m +1)+1 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m +1)+1 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w j + m +1)+1 → j + m +1)+2 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m +1)+2 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s ; κ w j + m + s +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; κ e j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 4 s + ( j + 1) s ; − κ w j + m +1) → j + m +1)+3 ⊗ e j +3 , i = j + s ;0 , otherwise,where κ = − κ ℓ ( α j + m +6) ) . (3) If r = 2 , then Ω ( Y (22) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = e j + m + s )+1 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w j + m ) − → j + m + s )+1 ⊗ e j , i = j − s ; e j + m + s )+1 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 .
36 MARIYA KACHALOVA If s j < s , then b ij = κ w j + m + s )+1 → j + m )+3 ⊗ e j +2 , i = j − s ; − κ e j + m )+3 ⊗ w j +2 → j +1) , i = ( j + 1) s , j < s − or j = 6 s − , otherwise,where κ = κ ℓ ( α j + m +5) ) f ( j, s ) .If s j < s , then b ij = − κ ℓ +1 ( α j + m ) ) e j + m +1) ⊗ e j +3 , i = j − s ;0 , otherwise. (4) If r = 3 , then Ω ( Y (22) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m )+2 → j + m )+3 ⊗ e j , i = j ; κ e j + m )+3 ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise,where κ = κ ℓ ( α j + m +5) ) .If s j < s , then b ij = − f (( j ) s , s − f ( j, s ) e j + m +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + 1) s ; w j + m +1) → j + m +1)+2 ⊗ w j + s )+1 → j + s )+2 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ e j + s )+2 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κ ℓ +1 ( α j + m +1) ) w j + m + s +1)+2 → j + m +2) ⊗ w j +3 → j +1) ,i = ( j + 1) s , j < s − − κ ℓ +1 ( α j + m +1) ) w j + m +1)+3 → j + m +2) ⊗ w j +3 → j + s +1)+1 ,i = 2 s + ( j + 1) s , j < s − κ ℓ +1 ( α j + m +1) ) e j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 4 s + ( j + 1) s ; − κ ℓ +1 ( α j + m +1) ) w j + m + s +1)+1 → j + m +2) ⊗ e j +3 , i = j + s ;0 , otherwise. (5) If r = 4 , then Ω ( Y (22) t ) is described with (8 s × s ) -matrix with the following elements b ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 137 If j < s , then b ij = − w j + m ) → j + m + s )+2 ⊗ e j , i = j + s ; − w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +1 , i = j + 2 s ; e j + m + s )+2 ⊗ w j → j +2 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = − w j + m ) − → j + m + s )+2 ⊗ e j , i = j − s ; − w j + m ) → j + m + s )+2 ⊗ e j , i = j ; w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +1 , i = j + 2 s ; − e j + m + s )+2 ⊗ w j → j +2 , i = j + 6 s ;0 , otherwise.If s j < s , then b ij = − κ ℓ +1 ( α j + m − ) e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − or j = 4 s − , otherwise.If s j < s , then b ij = − κ w j + m )+1 → j + m +1) ⊗ e j +2 , i = j ; κ e j + m +1) ⊗ w j +2 → j +1) , i = s + ( j + 1) s , j < s − , otherwise,where κ = − κ ℓ +1 ( α j + m ) ) .If s j < s , then b ij = − κ w j + m + s )+2 → j + m +1) ⊗ e j +2 , i = j + 2 s ; κ w j + m )+3 → j + m +1) ⊗ w j +2 → j +1) , i = ( j + 1) s , j = 6 s − κ e j + m +1) ⊗ w j +2 → j +1) , i = s + ( j + 1) s , j = 6 s − , otherwise,where κ = − κ ℓ +1 ( α j + m ) ) .If s j < s − , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ;0 , otherwise.If s − j < s , then b ij = 0 .
38 MARIYA KACHALOVA If s j < s , then b ij = − w j + m )+3 → j + m + s +1)+1 ⊗ e j +3 , i = j + s ; w j + m +1) → j + m + s +1)+1 ⊗ w j +3 → j +1) , i = s + ( j + 1) s , j = 8 s − , otherwise. (6) If r = 5 , then Ω ( Y (22) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ w j + m )+1 → j + m )+3 ⊗ e j , i = j ; κ w j + m + s )+1 → j + m )+3 ⊗ e j , i = j + s ; − κ e j + m )+3 ⊗ w j → j +2 , i = j + 4 s ; − κ e j + m )+3 ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise,where κ = κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = − κ w j + m + s )+1 → j + m +1) ⊗ e j , i = j − s ; κ e j + m +1) ⊗ w j → j + s )+1 , i = j + s ; − κ e j + m +1) ⊗ w j → j +1 , i = j + 2 s ;0 , otherwise,where κ = − κ ℓ +1 ( α j + m ) ) .If s j < s , then b ij = e j + m +1)+1 ⊗ w j +1 → j +1) , i = ( j + 1) s ; w j + m )+3 → j + m +1)+1 ⊗ w j +1 → j +2 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e j + m +1)+1 ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+2 , i = j − s ;0 , otherwise.If s j < s , then b ij = − w j + m )+3 → j + m +1)+1 ⊗ e j + s )+2 , i = j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 139 If s j < s , then b ij = 0 .If s j < s , then b ij = κ w j + m +1)+2 → j + m +1)+3 ⊗ e j +3 , i = j − s ; κ w j + m +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = ( j + 1) s , j < s − κ w j + m + s +1)+1 → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s , j < s − − κ e j + m +1)+3 ⊗ w j +3 → j +1)+2 , i = 4 s + ( j + 1) s , j < s − − κ e j + m +1)+3 ⊗ w j +3 → j + s +1)+2 , i = 5 s + ( j + 1) s , j < s − , otherwise,where κ = κ ℓ +1 ( α j + m ) ) . (7) If r = 6 , then Ω ( Y (22) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) → j + m + s )+1 ⊗ e j , i = ( j ) s ; e j + m + s )+1 ⊗ w j → j + s )+1 , i = j + 3 sf ( j, s );0 , otherwise.If s j < s , then b ij = κ ℓ +1 ( α j + m ) ) e j + m +1) ⊗ w j +1 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = κ w j + m )+2 → j + m )+3 ⊗ e j +2 , i = j + s ; − κ e j + m )+3 ⊗ w j +2 → j +3 , i = j + 2 s, j > s ;0 , otherwise,where κ = − κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = 0 .If s j < s , then b ij = w j + m )+3 → j + m + s +1)+2 ⊗ e j +3 , i = j ;0 , otherwise. (8) If r = 7 , then Ω ( Y (22) t ) is described with (9 s × s ) -matrix with the following elements b ij :
40 MARIYA KACHALOVA If j < s , then b ij = κ w j + m + s )+2 → j + m +1) ⊗ e j , i = j + s ; − κ w j + m )+3 → j + m +1) ⊗ w j → j +1 , i = j + 2 s ; κ e j + m +1) ⊗ w j → j +2 , i = j + 4 s ; κ e j + m +1) ⊗ w j → j + s )+2 , i = j + 5 s ;0 , otherwise,where κ = − κ ℓ +1 ( α j + m ) ) .If s j < s , then b ij = 0 .If s j < s , then b ij = − e j + m +1)+2 ⊗ w j +1 → j +1) , i = ( j + 1) s ; − w j + m )+3 → j + m +1)+2 ⊗ e j +1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − e j + m + s +1)+2 ⊗ w j + s )+1 → j +1) , i = ( j + s + 1) s ; − w j + m )+3 → j + m + s +1)+2 ⊗ e j + s )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κ w j + m +1+ sf ( j, s − → j + m +1)+3 ⊗ w j +3 → j +1) , i = s + ( j + 1) s ; κ e j + m +1)+3 ⊗ w j +3 → j + s +1+ sf ( j, s − , i = 2 s + ( j + 1) s ; κ w j + m + s +1)+1 → j + m +1)+3 ⊗ e j +3 , i = j − s ; κ w j + m +1)+1 → j + m +1)+3 ⊗ e j +3 , i = j ;0 , otherwise,where κ = − κ ℓ +1 ( α j + m ) ) .If s j < s , then b ij = 0 . (9) If r = 8 , then Ω ( Y (22) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w j + m ) − → j + m + s )+2 ⊗ e j , i = j, j < s ; − e j + m + s )+2 ⊗ w j → j + s )+1 , i = s + ( j + s ) s ; − w j + m + s )+1 → j + m + s )+2 ⊗ w j → j +2 , i = j + 3 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 141 If s j < s , then b ij = − κ w j + m )+2 → j + m )+3 ⊗ e j +1 , i = j − s ; κ w j + m + s )+1 → j + m )+3 ⊗ w j +1 → j +2 , i = j + s ; κ e j + m )+3 ⊗ w j +1 → j +1) , i = ( j + 1) s , j < s − or j = 4 s − , otherwise,where κ = − κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = 0 .If s j < s , then b ij = κ ℓ +1 ( α j + m ) ) e j + m +1) ⊗ w j +2 → j +3 , i = j ;0 , otherwise.If s j < s , then b ij = − w j + m +1) → j + m + s +1)+1 ⊗ e j +3 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (10) If r = 9 , then Ω ( Y (22) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κ ℓ +1 ( α j + m − ) e j + m )+3 ⊗ e j , i = j ;0 , otherwise.If s j < s , then b ij = w j + m +1) → j + m +1)+2 ⊗ e j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κ e j + m +2) ⊗ w j +3 → j + s +1+ sf ( j, s − , i = s + ( j + 1) s ; κ w j + m + s +1)+2 → j + m +2) ⊗ e j +3 , i = j ; − κ w j + m +1)+3 → j + m +2) ⊗ w j +3 → j +1) , i = ( j + 1) s , j = 6 s − , otherwise,where κ = − κ ℓ +1 ( α j + m +1) ) . (11) If r = 10 , then Ω ( Y (22) t ) is described with (7 s × s ) -matrix with the following elements b ij :
42 MARIYA KACHALOVA If j < s , then b ij = − κ w j + m ) → j + m )+3 ⊗ e j , i = j ; − κ w j + m + s )+1 → j + m )+3 ⊗ w j → j +1 , i = j + s ; κ w j + m )+1 → j + m )+3 ⊗ w j → j + s )+1 , i = j + 2 s ; κ w j + m + s )+2 → j + m )+3 ⊗ w j → j +2 , i = j + 3 s ; − κ w j + m )+2 → j + m )+3 ⊗ w j → j + s )+2 , i = j + 4 s ; κ e j + m )+3 ⊗ w j → j +3 , i = j + 5 s ;0 , otherwise,where κ = − κ ℓ +1 ( α j + m − ) .If s j < s , then b ij = − κ w j + m )+1 → j + m +1) ⊗ e j + s )+1 , i = j ; κ w j + m )+2 → j + m +1) ⊗ w j + s )+1 → j + s )+2 , i = j + 2 s ; κ w j + m )+3 → j + m +1) ⊗ w j + s )+1 → j +3 , i = j + 4 s, j < s ; κ e j + m +1) ⊗ w j + s )+1 → j +1) , i = ( j + 1) s , j < s − or j = 3 s − , otherwise,where κ = − κ ℓ +1 ( α j + m ) ) .If s j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s , j < s − w j + m )+3 → j + m + s +1)+1 ⊗ w j + s )+2 → j +3 , i = j + 2 s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = w j + m +1) → j + m + s +1)+1 ⊗ w j + s )+2 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s − , then b ij = 0 .If s − j < s , then b ij = − w j + m +1) → j + m +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − w j + m )+3 → j + m +1)+2 ⊗ e j +3 , i = j − s ; − w j + m +1) → j + m +1)+2 ⊗ w j +3 → j +1) , i = ( j + 1) s , j < s − , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 143 (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (22) t ) is a Ω r ( Y (22) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 39 (Translates for the case 23) . (I) Let r ∈ N , r < . r -translates of theelements Y (23) t are described by the following way. (1) If r = 0 , then Ω ( Y (23) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, = w j + m ) → j + m +1) ⊗ e j . (2) If r = 1 , then Ω ( Y (23) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b j, s = w j + m +1) → j + m +2) ⊗ e j +3 . (3) If r = 2 , then Ω ( Y (23) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, s = w j + m +1) → j + m +2) ⊗ e j +3 . (4) If r = 3 , then Ω ( Y (23) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, s = w j + m +1) → j + m +2) ⊗ e j +2 ; b j, s = w j + m +1) → j + m +2) ⊗ e j +2 . (5) If r = 4 , then Ω ( Y (23) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b j,s = w j + m ) → j + m +1) ⊗ e j . (6) If r = 5 , then Ω ( Y (23) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, s = w j + m +1) → j + m +2) ⊗ e j +1 ; b j, s = w j + m +1) → j + m +2) ⊗ e j +1 . (7) If r = 6 , then Ω ( Y (23) t ) is described with (9 s × s ) -matrix with the following two nonzeroelements : b j, = w j + m ) → j + m +1) ⊗ e j ; b j, s = w j + m +1) → j + m +2) ⊗ e j +3 . (8) If r = 7 , then Ω ( Y (23) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, s = w j + m +1) → j + m +2) ⊗ e j +2 ; b j, s = w j + m +1) → j + m +2) ⊗ e j +2 . (9) If r = 8 , then Ω ( Y (23) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, s = w j + m +1) → j + m +2) ⊗ e j +3 . (10) If r = 9 , then Ω ( Y (23) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b j,s = w j + m +1) → j + m +2) ⊗ e j + s )+1 ; b j, s = w j + m +1) → j + m +2) ⊗ e j + s )+1 .
44 MARIYA KACHALOVA (11) If r = 10 , then Ω ( Y (23) t ) is described with (6 s × s ) -matrix with one nonzero elementthat is of the following form : b j, = w j + m ) → j + m +1) ⊗ e j . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (23) t ) is a Ω r ( Y (23) t ) , whose left components twisted by σ ℓ . Proposition 40 (Translates for the case 24) . (I) Let r ∈ N , r < . r -translates of theelements Y (24) t are described by the following way. (1) If r = 0 , then Ω ( Y (24) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, = w j +3 → j +1)+3 ⊗ e j +3 . (2) If r = 1 , then Ω ( Y (24) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b j, s = w j + m )+3 → j + m +1)+3 ⊗ e j +2 ; b j, s = w j + m )+3 → j + m +1)+3 ⊗ e j +2 . (3) If r = 2 , then Ω ( Y (24) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, = w j + m )+3 → j + m +1)+3 ⊗ e j . (4) If r = 3 , then Ω ( Y (24) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j − s, s = w j + m )+2 → j + m )+3 ⊗ w j +1 → j ; b j − s, s = − w j + m )+2 → j + m )+3 ⊗ w j +1 → j . (5) If r = 4 , then Ω ( Y (24) t ) is described with (9 s × s ) -matrix with the following two nonzeroelements : b j, = e j + m )+3 ⊗ w j → j +1) ; b j,s = e j + m ) ⊗ w j → j +1) . (6) If r = 5 , then Ω ( Y (24) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, = e j + m )+1 ⊗ w j → j +1) ; b j,s = e j + m )+1 ⊗ w j → j +1) . (7) If r = 6 , then Ω ( Y (24) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b j, = e j + m ) ⊗ w j → j +1) . (8) If r = 7 , then Ω ( Y (24) t ) is described with (8 s × s ) -matrix with the following two nonzeroelements : b j, = e j + m )+2 ⊗ w j → j +1) ; b j,s = e j + m )+2 ⊗ w j → j +1) . (9) If r = 8 , then Ω ( Y (24) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b j, = e j + m )+3 ⊗ w j → j +1) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E . II. 145 (10) If r = 9 , then Ω ( Y (24) t ) is described with (7 s × s ) -matrix with the following two nonzeroelements : b j + s, = − w j + m ) → j + m )+3 ⊗ w j → j +1 ; b j +2 s, = w j + m ) → j + m )+3 ⊗ w j → j +1)+1 . (11) If r = 10 , then Ω ( Y (24) t ) is described with (6 s × s ) -matrix with the following nonzeroelements : b j, = − w j + m ) → j + m +1) ⊗ e j ; b j − s, s = w j + m ) → j + m )+3 ⊗ w j +3 → j ; b j, s = w j + m )+3 → j + m +1)+3 ⊗ e j +3 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (24) t ) is a Ω r ( Y (24) t ) , whose left components twisted by σ ℓ . Multiplications in HH ∗ ( R )From the descriptions of elements Y ( i ) t and its Ω-translates we can find multiplications of theelements using the formula ( ∗ ).We will find a multiplication of elements of the types 4 and 3 for s > Y (4) t and Y (3) t . For its degrees t and t we have: t = 11 ℓ + 1 , ℓ ( n + s ) ≡ s (2 s ) , ℓ ... 2 or char K = 2; t = 11 ℓ + 1 , ℓ ( n + s ) ≡ s ) , ℓ ... 2 or char K = 2 . Let t = t + t ; this is the degree of an element Y (4) t Y (3) t . Then t = 11( ℓ + ℓ ) + 2. Group of thedegree t has type (5). Y (3) t is an (7 s × s )-matrix with two nonzero elements y , = w → ⊗ e and y ,s = w → s +1 ⊗ e . Ω t ( Y (4) t ) is an (6 s × s )-matrix that was described in proposition 20.Multiplication of Ω t ( Y (4) t ) and Y (3) t an (6 s × s )-matrix with the following two nonzero elements: b , = κ ℓ ( α ) κ ℓ ( α ) κ ℓ ( α ) w → ⊗ e ; b , s − = − κ ℓ ( α s − ) w → s +2 ⊗ w s − → . We have ℓ ... 2, hence κ ℓ ( α ) = κ ℓ ( α s − ) = 1, κ ℓ ( α ) = κ ℓ ( α ), κ ℓ ( α ) κ ℓ ( α ) κ ℓ ( α ) = κ ℓ ( α ) κ ℓ ( α ) = κ ℓ + ℓ ( α ). Y (5) t is an (6 s × s )-matrix with a single nonzero element y , = κ ℓ ( α ) w → ⊗ e , hence Y (4) Y (3) coincide with Y (5) for degree of type (5).Multiplications of other elements, except Y (5) , Y (10) , Y (17) , Y (19) and Y (21) , are similarlyconsidered. To get the whole picture we should prove the following lemma. Lemma 41. ( a ) Let Y (5) be an arbitrary element from generators of the corresponding type. Then there areelements Y (3) and Y (4) such as Y (5) = Y (3) Y (4) . ( b ) Let Y (10) be an arbitrary element from generators of the corresponding type. Then thereare elements Y (3) and Y (6) such as Y (10) = Y (3) Y (6) . ( c ) Let Y (17) be an arbitrary element from generators of the corresponding type. Then thereare elements Y (3) and Y (15) such as Y (17) = Y (3) Y (15) . ( d ) Let Y (19) be an arbitrary element from generators of the corresponding type. Then thereare elements Y (3) and Y (18) such as Y (19) = Y (3) Y (18) .
46 MARIYA KACHALOVA ( e ) Let Y (21) be an arbitrary element from generators of the corresponding type. Then thereare elements Y (3) and Y (20) such as Y (21) = Y (3) Y (20) .Proof. The degree 1 has type 3, for all s . It only remains to use the relations for type (3). (cid:3) References [1] C. Riedtmann,
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