Hochschild cohomology ring for self-injective algebras of tree class E 6
aa r X i v : . [ m a t h . K T ] N ov HOCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRASOF TREE CLASS E MARIYA PUSTOVYKH
Abstract.
We describe the Hochschild cohomology ring for one of the two self-injective algebrasof tree class E in terms of generators and relations. 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 ) 135References 1391. 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 ). Consider algebras of tree class E . Any algebra of the class E is derived equivalent to thepath 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 − β , αγβ , βγα .2) 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 − β , α t γ t − β t − , β t γ t − α t − (1 t s − α γ s − α s − , β γ s − β s − .Henceforth we will often omit indexes in arrows α i , β i and γ i as long as subscripts are clearfrom the context.The present paper is dedicated to the study of Hochschild cohomology ring structure for algebra R s . For this algebra we obtain the description of Hochschild cohomology ring structure in termsof generators and relations. In studies of the structure of cohomology ring we will construct thebimodule resolution of R s , which could be seen as a whole 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.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 ... 2, ℓn + m ≡ s ) or s = 1;(2) r = 0, ℓ ... 2, char K = 3, ℓn + m ≡ s ) or s = 1;(3) r = 1, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(4) r = 1, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(5) r = 2, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(6) r = 3, ℓ ... 2, char K = 2, ℓn + m ≡ s ) or s = 1;(7) r = 3, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(8) r = 4, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(9) r = 4, ℓ ... 2, char K = 3, ℓn + m ≡ s ) or s = 1;(10) r = 4, ℓ ... 2, char K = 2, ℓn + m ≡ s ) or s = 1;(11) r = 5, ℓ ... 2, char K = 3, ℓn + m ≡ s ) or s = 1;(12) r = 5, ℓ ... 2, char K = 3, ℓn + m ≡ s ) or s = 1;(13) r = 6, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(14) r = 6, ℓ ... 2, char K = 3, ℓn + m ≡ s ) or s = 1;(15) r = 6, ℓ ... 2, char K = 2, ℓn + m ≡ s ) or s = 1;(16) r = 7, ℓ ... 2, ℓn + m ≡ s ) or s = 1; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (17) r = 7, ℓ ... 2, char K = 2, ℓn + m ≡ s ) or s = 1;(18) r = 8, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(19) r = 9, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(20) r = 9, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(21) r = 10, ℓ ... 2, ℓn + m ≡ s ) or s = 1;(22) r = 10, ℓ ... 2, char K = 3, ℓn + m ≡ s ) or s = 1.Let M = 11 s gcd(2 n,s ) . 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. 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 TT 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 (1) = X (3) X (2) = X (3) X (3) = X (3) X (5) = X (3) X (8) = 0; X (3) X (9) = X (3) X (10) = X (3) X (11) = X (3) X (12) = X (3) X (14) = 0; X (3) X (16) = X (3) X (17) = X (3) X (19) = X (3) X (21) = X (3) X (22) = 0; X (3) X (4) = e X (5) ; X (3) X (6) = e X (10) ; X (3) X (7) = 2 e X (8) ; X (3) X (13) = 2 e X (16) ; X (3) X (15) = e X (17) ; X (3) X (18) = e X (19) ; X (3) X (20) = e X (21) . MARIYA PUSTOVYKH X (4) X (7) = ( e X (10) , char K = 2 , , otherwise; (r1) X (7) X (7) = ( s e X (14) , char K = 3 , , otherwise; (r2) X (4) X (13) = ( e X (17) , char K = 2 , , otherwise; (r3) X (7) X (18) = ( − s e X (2) , char K = 3 , , otherwise; (r4) X (13) X (20) = ( e X (10) , char K = 2 , , otherwise; (r5) X (18) X (18) = ( − 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). X (1) X (2) X (4) X (6) X (7) X (8) X (9) X (11) X (12) X (1) e X (1) e X (2) e X (4) e X (6) e X (7) e X (8) e X (9) e X (11) e X (12) X (2) e X (8) X (4) e X (8) (1) 0 e X (11) s e X (14) X (6) X (7) (2) 0 s e X (16) X (8) X (9) − s e X (19) X (11) − s e X (21) X (13) X (14) X (15) X (16) X (18) X (20) X (22) X (1) e X (13) e X (14) e X (15) e X (16) e X (18) e X (20) e X (22) X (2) e X (14) e X (21) X (4) (3) 0 e X (16) e X (20) X (6) e X (19) e X (20) X (7) − e X (20) e X (19) − e X (21) (4) 0 − s e X (5) X (8) − e X (21) X (9) − e X (22) − e X (21) s e X (3) s e X (5) X (11) s e X (5) OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E X (12) X (13) X (14) X (15) X (16) X (18) X (20) X (22) X (12) − s e X (2) s e X (8) X (13) e X (4) e X (3) e X (5) − e X (7) (5) e X (11) X (14) X (15) e X (4) e X (6) e X (8) X (16) − e X (8) X (18) (6) (7) s e X (16) 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 , X (25)0 , X (26)0 , X (27)0 , X (28)0 o , char K = 3; X ∪ n X (24)0 , X (25)0 , X (26)0 , X (27)0 , X (28)0 o , char K = 3;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 = deg X (25)0 = deg X (26)0 = deg X (27)0 = deg X (28)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 ( i ) = e X ( i ) , t = 0;0 , otherwise , i ∈ { , , } ; X (1) X ( i ) = e X ( i ) , t = 0; e X (2) , t > K = 3;0 , otherwise , i ∈ { , , } ; X (9) X ( i ) = e X (8) , i ∈ { , , } ; X (13) X ( i ) = e X (14) , char K = 3;0 , otherwise , i ∈ { , , } ; X (22) X ( i ) = e X (21) , i ∈ { , , } ; X ( j ) X ( i ) =0 , j ∈ [2 , \ { , , } , i ∈ [23 , , where t denotes a degree of the element X (1) . MARIYA PUSTOVYKH
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 ns = { , , . . . , ns − } , be the idempotents of the algebra K [ Q s ], thatcorrespond 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 ns . 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 − R = R s . We introduce an automorphism σ : R → R , which is mapping as follows: σ ( e i ) = e i + n , i ≡ , n ); e i + n +2 , i ≡ , n ); e i + n − , i ≡ , n ) ,σ ( γ i ) = − γ i + n ,σ ( α i ) = − β i +3 n , i ≡ , β i +3 n , i ≡ , σ ( β i ) = − α i +3 n , i ≡ , α i +3 n , i ≡ . Define the helper functions f : Z × Z → Z , h : Z × Z → Z and λ : Z → Z , which act in thefollowing 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,λ ( i ) = i, i ≡ , i + 2 , i ≡ , i − , i ≡ , . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E 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 ) n − h ( m, i,rn ⊕ f ( m, M i =0 (cid:0) P ( r + m ) n +1+( m ) +3 i,rn +1 ⊕ P λ (( r + m ) n +1+( m ) +3 i ) ,rn +3 (cid:1) ⊕ f ( m, M i =0 (cid:0) P ( r + m ) n +1+( m +1) +3 f ( m, i,rn +2 ⊕ P λ (( r + m ) n +1+( m +1) +3 f ( m, i ) ,rn +4 (cid:1) ⊕ f ( m, M i =0 P ( r + m +1) n − h ( m, i,rn +5 ,Q ′ m +1 ,r = − f ( m, M i =0 P ( r + m ) n +1+ h ( m, f ( m, i,rn ⊕ P ( r + m +1) n − h ( m, − f ( m, ,rn +1 ⊕ P λ (( r + m +1) n − h ( m, − f ( m, ,rn +3 ⊕ P ( r + m +1) n − h ( m, f ( m, ,rn +2 ⊕ P λ (( r + m +1) n − h ( m, f ( m, ,rn +2) ,rn +4 ⊕ − f ( m, M i =0 P ( r + m +1) n +1+ h ( m, − f ( m, i,rn +5 . 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) Denote by w ( m ) i → the way that starts in i th vertex and has length m .(3) Denote by w ( m ) → i the way that ends in i th vertex and has length m . MARIYA PUSTOVYKH
Define the helper functions g : Z → Z f : Z × Z → Z and f : Z × Z → Z , which act in thefollowing way: g ( j ) = , ( j ) s < s ;3 otherwise , f ( x, y ) = , x < y ;0 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 ) n → ( j + m ) n + g ( j ) ⊗ e jn , i = ( j ) s ; − e ( j + m ) n + g ( j ) ⊗ w jn → jn + g ( j ) , i = j + s ;0 otherwise.If 2 s j < s , then( d ) ij = w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j ) , i = j − s ; − e ( j + m ) n + g ( j )+1 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ;0 otherwise.If 4 s j < s , then( d ) ij = w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ; − e ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = 5 s + ( j ) s ;0 otherwise.If 6 s j < s , then( d ) ij = w ( j + m ) n +5 → ( j + m ) n +6 ⊗ e jn +5 , i = j − s ; − e ( j + m ) n +6 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ;0 otherwise. Description of the d d : Q → Q – is an (6 s × s ) matrix . If 0 j < s , then( d ) ij = w jn +1+ j +2 f ( j , → jn +5 ⊗ w jn → jn + j , i = j + 2 sj , j < − w jn +3+ j → jn +5 ⊗ w jn → jn + j +2(1 − f ( j , , i = j + 2 sj + s, j < OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then( d ) ij = w (4 − j ) → ( j +1) n +1+ g ( j + s ) ⊗ w ( j ) jn + g ( j + s ) → ,i = j + 2 sj + s − s (1 − f ( j, s )) f ( j , , j < w (1 − j ) → ( j +1) n +1+ g ( j + s ) ⊗ w ( j +3) jn + g ( j + s ) → ,i = ( j + 1) s + 2 sj + s (1 − f ( j, s )) , j < s j < s , then( d ) ij = w (2) → ( j +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ; w (1) → ( j +1) n + g ( j ) ⊗ w (1) jn + g ( j + s )+1 → , i = ( j ) s + 6 s ; e ( j +1) n + g ( j ) ⊗ w (2) jn + g ( j + s )+1 → , i = ( j + 1) s + sf ( j, s );0 otherwise.If 5 s j < s , then( d ) ij = w ( j +1) n → ( j +2) n ⊗ e jn +5 , i = j + s ; w ( j +1) n +1+ j +2 f ( j , f ( j , → ( j +2) n ⊗ w jn +5 → ( j +1) n + j +2 f ( j , ,i = ( j + 1) s + 2 sj , j < Description of the d d : Q → Q – is an (8 s × s ) matrix . If 0 j < s , then( d ) ij = w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ e jn , i = ( j ) s ; − f ( j, s ) e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ; f ( j, s ) w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn +1+ g ( j + s ) , i = ( j + s ) s + 3 s ;0 otherwise.If 2 s j < s , then( d ) ij = w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ; − w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ; − f ( j, s ) e ( j + m ) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 otherwise. If 4 s j < s , then( d ) ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ; − e ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = 5 s + ( j ) s ; w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s , j < s ;0 otherwise.If 6 s j < s , then( d ) ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = 5 s + ( j ) s ; − w ( j + m +1) n − → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s , j < s ; − e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = 3 s + ( j + 1) s + sf ( j, s );0 otherwise. Description of the d d : Q → Q – is an (9 s × s ) matrix . If 0 j < s , then ( d ) ij = w ( j + m ) n +2 → ( j + m ) n +5 ⊗ e jn , i = j ; − w ( j + m ) n +4 → ( j + m ) n +5 ⊗ e jn , i = j + s ; e ( j + m ) n +5 ⊗ w jn → jn +1 , i = j + 2 s ; e ( j + m ) n +5 ⊗ w jn → jn +3 , i = j + 3 s ;0 otherwise.If s j < s , then( d ) ij = w ( j + m ) n +2+ j +2(1 − f ( j , → ( j + m +1) n ⊗ w jn → jn + j , i = j − s + 2 sj , j < − e ( j + m +1) n ⊗ w jn → jn +4 , i = j + 4 s ;0 otherwise.If 2 s j < s , then( d ) ij = w (2 − j ) → ( j + m +1) n + g ( j + s ) ⊗ w ( j ) jn + g ( j ) → , i = 2 j s + j, j < e ( j + m +1) n + g ( j + s ) ⊗ w (2) jn + g ( j ) → , i = 6 s + ( j + s ) s ;0 otherwise.If 4 s j < s , then( d ) ij = w (1 − j ) → ( j + m +1) n + g ( j ) ⊗ w ( j ) jn + g ( j )+1 → , i = j + 2 sj , j < OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If 6 s j < s , then( d ) ij = w (2) → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 , i = j − s ; w (1) → ( j + m +1) n + g ( j + s )+1 ⊗ w (1) jn + g ( j )+1 → , i = ( j + s ) s + 6 s ; − f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w (2) jn + g ( j )+1 → , i = ( j + 1) s + sf ( j, s );0 otherwise.If 8 s j < s , then( d ) ij = w ( j + m +1) n +2 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; e ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +1 , i = 2 s + ( j + 1) s ; w ( j + m +1) n +1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ; − w ( j + m +1) n +3 → ( j + m +1) n +5 ⊗ e jn +5 , 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 − n +5 → ( j + m ) n +1 ⊗ e jn , i = j, j < s ; − f ( j, s ) w ( j + m ) n → ( j + m ) n + g ( j ) ⊗ e jn , i = ( j ) s + s ; − e ( j + m ) n + g ( j ) ⊗ w jn → jn + g ( j + s ) , i = ( j + s ) s + 2 s ; e ( j + m ) n + g ( j ) ⊗ w jn → jn +1+ g ( j ) , i = j + 4 s ;0 otherwise.If 2 s j < s , then( d ) ij = w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn +1 → ( j +1) n , i = ( j + 1) s , j < s ; w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j ; − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = 4 s + j ; − f ( j, s ) e ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s + s ;0 otherwise.If 4 s j < s , then( d ) ij = e ( j + m ) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s , j < s ; w ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j ; − w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j + 2 s ; − f ( j, s ) e ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = 8 s + ( j ) s ;0 otherwise. If 6 s j < s , then( d ) ij = − w ( j + m +1) n − → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s , j < s ; w ( j + m +1) n − → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , i = 8 s + ( j ) s ; f ( j, s ) w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) ,i = 2 s + ( j + 1) s + sf ( j, s ); − f ( j, s ) e ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 ,i = 6 s + ( j + 1) s + sf ( j, s );0 otherwise. Description of the d d : Q → Q – is an (9 s × s ) matrix . If 0 j < s , then( d ) ij = w ( j + m ) n +1 → ( j + m +1) n ⊗ e jn , i = j ; w ( j + m ) n +3 → ( j + m +1) n ⊗ e jn , i = j + s ;(2 f ( j , − w ( j + m +1) n − j → ( j + m +1) n ⊗ w jn → jn +1+ j ,i = j + 2 s (1 + j ) , j < f ( j , − w ( j + m +1) n − j → ( j + m +1) n ⊗ w jn → jn +3+ j ,i = j + 2 s (1 + j ) + s, j < s j < s , then( d ) ij = w (1) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ; − e ( j + m +1) n + g ( j + s ) ⊗ w (3) jn + g ( j + s ) → , i = ( j + 1) s + s (1 − f ( j, s ));0 otherwise.If 3 s j < s , then( d ) ij = − w (3) → ( j + m +1) n + g ( j )+1 ⊗ w (1) jn + g ( j + s ) → , i = j + s ; w (2) → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j − s ; w (1) → ( j + m +1) n + g ( j )+1 ⊗ w (3) jn + g ( j + s ) → , i = ( j + 1) s + sf ( j, s ); − f ( j, s ) e ( j + m +1) n + g ( j )+1 ⊗ w (2) jn + g ( j + s ) → , i = ( j ) s + 6 s ;0 otherwise.If 5 s j < s , then( d ) ij = f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w (1) jn + g ( j + s )+1 → , i = j + s ; w (3) → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If 7 s j < s , then( d ) ij = w ( j + m +1) n +1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n +3 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = s + ( j + 1) s ; − e ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +2 , i = 4 s + ( j + 1) s ; − e ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +4 , i = 5 s + ( j + 1) s ; w ( j + m +1) n +2 → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ; − w ( j + m +1) n +4 → ( j + m +1) n +5 ⊗ e jn +5 , i = j ;0 otherwise.If 8 s j < s , then( d ) ij = w ( j + m +1) n +2 → ( j + m +2) n ⊗ e jn +5 , i = j − s ; − e ( j + m +2) n ⊗ w jn +5 → ( j +1) n +3 , i = 3 s + ( j + 1) 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 ) n → ( j + m ) n + g ( j )+1 ⊗ e jn , i = ( j ) s ; − w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ; − e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = ( j + s ) s + 3 s ; e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 5 s ;0 otherwise.If 2 s j < s , then( d ) ij = w ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ; − w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j + s ; − w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 3 s ; e ( j + m ) n +5 ⊗ w jn + g ( j ) → jn +5 , i = ( j ) s + 7 s ;0 otherwise.If 4 s j < s , then( d ) ij = w ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j + s ; − f ( j, s ) e ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = ( j ) s + 8 s ; e ( j + m +1) n ⊗ w jn +4 → ( j +1) n , i = ( j + 1) s , j > s ; − w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn +4 → jn +5 , i = j + 2 s, j > s ;0 otherwise. If 6 s j < s , then( d ) ij = w ( j + m ) n +5 → ( j + m +1) n +1 ⊗ e jn +5 , i = j + s, j < s ; − w ( j + m +1) n → ( j + m +1) n +1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s , j < s ; − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = ( j ) s + 8 s ; e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + s + s (1 − f ( 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 ) n +2 → ( j + m ) n +5 ⊗ e jn , i = j ; − w ( j + m ) n +4 → ( j + m ) n +5 ⊗ e jn , i = j + s ; e ( j + m ) n +5 ⊗ w jn → jn +1 , i = j + 2 s ; − e ( j + m ) n +5 ⊗ w jn → jn +3 , i = j + 3 s ;0 otherwise.If s j < s , then( d ) ij = − e ( j + m +1) n + g ( j + s )+1 ⊗ w (3) jn + g ( j + s ) → , i = ( j + 1) s + s (1 − f ( j, s )); w (3) → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j + s ; w (2) → ( j + m +1) n + g ( j + s )+1 ⊗ w (1) jn + g ( j + s ) → , i = j + 3 s ; − w (1) → ( j + m +1) n + g ( j + s )+1 ⊗ w (2) jn + g ( j + s ) → , i = j + 5 s ;0 otherwise.If 3 s j < s , then( d ) ij = w (1) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ; e ( j + m +1) n + g ( j ) ⊗ w (1) jn + g ( j + s )+1 → , i = ( j ) s + 6 s ;0 otherwise.If 5 s j < s , then( d ) ij = w ( j + m +1) n +4 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ; − w ( j + m +1) n +5 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n +1 , i = ( j + 1) s + 2 s ; − e ( j + m +2) n ⊗ w jn +5 → ( j +1) n +2 , i = ( j + 1) s + 4 s ; − e ( j + m +2) n ⊗ w jn +5 → ( j +1) n +4 , i = ( j + 1) s + 5 s ; w ( j + m +1) n +1 → ( j + m +2) n ⊗ e jn +5 , i = j + s ; w ( j + m +1) n +3 → ( j + m +2) n ⊗ e jn +5 , i = j + 2 s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E Description of the d d : Q → Q – is an (7 s × s ) matrix . If 0 j < s , then( d ) ij = w ( j + m − n +5 → ( j + m ) n +5 ⊗ e jn , i = j ; − e ( j + m ) n +5 ⊗ w jn → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +2 → ( j + m ) n +5 ⊗ w jn → jn +1 , i = j + s ; w ( j + m ) n +4 → ( j + m ) n +5 ⊗ w jn → jn +3 , i = j + 2 s ; w ( j + m ) n +3 → ( j + m ) n +5 ⊗ w jn → jn +2 , i = j + 3 s ; − w ( j + m ) n +1 → ( j + m ) n +5 ⊗ w jn → jn +4 , i = j + 4 s ;0 otherwise.If s j < s , then( d ) ij = w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s , j < s ; w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ; − w ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 2 s ; e ( j + m +1) n ⊗ w jn + g ( j + s ) → jn +5 , i = ( j ) s + 5 s ;0 otherwise.If 3 s j < s , then( d ) ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s , j < s ; w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ; − e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 ,i = ( j + 1) s + 3 s + s (1 − f ( j, s )); − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 ,i = ( j ) s + 5 s ;0 otherwise.If 5 s j < s , then( d ) ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn +5 , i = ( j ) s + 5 s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s , j > s ; e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) ,i = ( j + 1) s + s + s (1 − f ( j, s )); w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j )+1 ,i = ( j + 1) s + 3 s + sf ( 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 ) n +5 → ( j + m +1) n ⊗ e jn , i = j ; e ( j + m +1) n ⊗ w jn → jn +1 , i = j + s ; − e ( j + m +1) n ⊗ w jn → jn +3 , i = j + 2 s ;0 otherwise.If s j < s , then( d ) ij = w (1) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j ; e ( j + m +1) n + g ( j ) ⊗ w (1) jn + g ( j + s ) → , i = j + 2 s ;0 otherwise.If 3 s j < s , then( d ) ij = w (1) → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j ; e ( j + m +1) n + g ( j )+1 ⊗ w (1) jn + g ( j + s )+1 → , i = ( j ) s + 5 s ;0 otherwise.If 5 s j < s , then( d ) ij = e ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n +2 → ( j + m +1) n +5 ⊗ e jn +5 , i = j ; − w ( j + m +1) n +4 → ( j + m +1) n +5 ⊗ e jn +5 , 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 ) n → ( j + m +1) n ⊗ e jn , i = j ; − e ( j + m +1) n ⊗ w jn → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +3 → ( j + m +1) n ⊗ w jn → jn +1 , i = j + s ; w ( j + m ) n +1 → ( j + m +1) n ⊗ w jn → jn +3 , i = j + 2 s ; w ( j + m ) n +4 → ( j + m +1) n ⊗ w jn → jn +2 , i = j + 3 s ; − w ( j + m ) n +2 → ( j + m +1) n ⊗ w jn → jn +4 , i = j + 4 s ; w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn +5 , i = j + 5 s ;0 otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then( d ) ij = f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j ; − e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s + s (1 − f ( j, s )); − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 2 s ; − f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → jn +5 , i = ( j ) s + 5 s ;0 otherwise.If 3 s j < s , then( d ) ij = − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j ; w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) ,i = ( j + 1) s + s + s (1 − f ( j, s )); − e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 ,i = ( j + 1) s + 3 s + s (1 − f ( j, s )); f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = ( j ) s + 5 s ;0 otherwise.If 5 s j < s , then( d ) ij = − w ( j + m +1) n → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn +5 , i = j ; w ( j + m +1) n +3 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +1 , i = ( j + 1) s + s ; − w ( j + m +1) n +1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +3 , i = ( j + 1) s + 2 s ; − w ( j + m +1) n +4 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +2 , i = ( j + 1) s + 3 s ; w ( j + m +1) n +2 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +4 , i = ( j + 1) s + 4 s ; − e ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +5 , 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 . 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 rn is of the form . . . −→ P ( r +3) n +5 ( α − β ) −→ P ( r +3) n +2 ⊕ P ( r +3) n +4 ( α β ) −→−→ P ( r +3) n ( γα γβ ) −→ P ( r +2) n +1 ⊕ P ( r +2) n +3 ( αγ − α β ) −→−→ P ( r +1) n +5 ⊕ P ( r +2) n ( α γα − β ) −→ P ( r +1) n +2 ⊕ P ( r +1) n +4 ( α γ β γ ) −→−→ P rn +5 ( α − β ) −→ P rn +1 ⊕ P rn +3 ( α β ) −→ P rn −→ S rn −→ . At that Ω ( S rn ) ≃ S ( r +4) n +5 . Lemma 5.
The begin of the minimal projective resolution of S rn +1 is of the form . . . −→ P rn +2 α −→ P rn +1 −→ S rn +1 −→ . At that Ω ( S rn +1 ) ≃ S ( r +1) n +2 . Lemma 6.
The begin of the minimal projective resolution of S rn +2 is of the form . . . −→ P ( r +4) n +3 β −→ P ( r +4) n γα −→ P ( r +3) n +2 α γ −→ P ( r +2) n +5 ( α − β ) −→−→ P ( r +2) n +1 ⊕ P ( r +2) n +4 ( α β ) −→ P ( r +2) n γβ −→−→ P ( r +1) n +3 βγ −→ P rn +5 α −→ P rn +2 −→ S rn +2 −→ . At that Ω ( S rn +2 ) ≃ S ( r +5) n +3 . Lemma 7.
The begin of the minimal projective resolution of S rn +3 is of the form . . . −→ P rn +4 β −→ P rn +3 −→ S rn +3 −→ . At that Ω ( S rn +3 ) ≃ S ( r +1) n +4 . Lemma 8.
The begin of the minimal projective resolution of S rn +4 is of the form . . . −→ P ( r +4) n +1 α −→ P ( r +4) n γβ −→ P ( r +3) n +4 β γ −→ P ( r +2) n +5 ( α − β ) −→−→ P ( r +2) n +2 ⊕ P ( r +2) n +3 ( α β ) −→ P ( r +2) n γα −→−→ P ( r +1) n +1 αγ −→ P rn +5 β −→ P rn +4 −→ S rn +4 −→ . At that Ω ( S rn +4 ) ≃ S ( r +5) n +1 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E Lemma 9.
The begin of the minimal projective resolution of S rn +5 is of the form . . . −→ P ( r +1) n γ −→ P rn +5 −→ S rn +5 −→ . At that Ω ( S rn +5 ) ≃ S ( r +2) n .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 [12]), as revised in [3]:
Lemma 10 (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 – 9 and Happel’slemma.As proved in [13], 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 11.
We have isomorphism Ω ( Λ R ) ≃ R σ . Proposition 12.
Automorphism σ has a finite order, and (1) if char K = 2 , then order of σ is equal to s gcd(2 n,s ) ; (2) if char K = 2 , then order of σ is equal to s gcd(2 n,s ) , if s gcd( n,s ) is divisible by , and to s gcd(2 n,s ) otherwise. Proposition 13.
The minimal period of bimodule resolution of R is s gcd(2 n,s ) .Proof. Since Ω ( Λ R ) ≃ R σ , period is equal to 11 a . There is an isomorphism σ a ( Q ) ≃ Q , i.e. σ a must act identically on idempotents. Hence, if char K = 2, then a = deg σ . For the casechar K = 2 we just need to show that a = deg σ , if s gcd( n,s ) is not divisible by 4. Consider thecase char K = 2, s gcd( n,s ) is not divisible by 4. Denote by b = deg σ . We see that σ b ( e i ) = e i ,σ b ( γ i ) = γ i , σ b ( α i ) = − α i , i ≡ , α i , i ≡ , σ b ( β i ) = − β i , i ≡ , β i , i ≡ . Let x = s − P i =0 e i +1 + e i +2 + e i +3 + e i +4 − e i − e i +5 . We have: x − = x , σ b ( w ) = xwx − , for anypath w , so σ b is an inner automorphism of R , and 11 b is a period of bimodule resolution. (cid:3) The additive structure of HH ∗ ( R ) Proposition 14 (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 t , R ) = s, ℓn + m ≡ s ) or ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) or ℓn + m ≡ s ) , ℓ ... , otherwise. (2) If r = 1 , then dim K Hom Λ ( Q t , R ) = s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (3) If r ∈ { , } , then dim K Hom Λ ( Q t , R ) = s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (4) If r ∈ { , , } , then dim K Hom Λ ( Q t , R ) = s, ℓn + m ≡ s );0 , otherwise. (5) If r = 4 , then dim K Hom Λ ( Q t , R ) = s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (6) If r = 6 , then dim K Hom Λ ( Q t , R ) = s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (7) If r = 9 , then dim K Hom Λ ( Q t , R ) = s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (8) If r = 10 , then dim K Hom Λ ( Q t , R ) = s, ℓn + m ≡ s ) or ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) or ℓn + m ≡ s ) , ℓ ... , 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 15 (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 = 0 , then dim K Hom Λ ( Q t , R ) = , ℓ ... , ℓ ... . (2) If r = 1 , then dim K Hom Λ ( Q t , R ) = , ℓ ... , ℓ ... . (3) If r ∈ { , } , then dim K Hom Λ ( Q t , R ) = 4 . (4) If r ∈ { , , } , then dim K Hom Λ ( Q t , R ) = 8 . (5) If r = 4 , then dim K Hom Λ ( Q t , R ) = , ℓ ... , ℓ ... . (6) If r = 6 , then dim K Hom Λ ( Q t , R ) = , ℓ ... , ℓ ... . (7) If r = 9 , then dim K Hom Λ ( Q t , R ) = , ℓ ... , ℓ ... . (8) If r = 10 , then dim K Hom Λ ( Q t , R ) = , ℓ ... , ℓ ... . Proof.
The proof is basically the same as proof of proposition 14. (cid:3)
Proposition 16 (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 , then dim K Im δ t = s − , ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (2) If r = 1 , then dim K Im δ t = s, ℓn + m ≡ s ) , ℓ ... s − , ℓn + m ≡ s ) , ℓ ... , otherwise. (3) If r = 2 , then dim K Im δ t = s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (4) If r = 3 , then dim K Im δ t = s − , ℓn + m ≡ s ) , ℓ ... , char K = 2;5 s, ℓn + m ≡ s ) , ℓ ... , char K = 2;7 s − , ℓn + m ≡ s ) , ℓ ... , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (5) If r = 4 , then dim K Im δ t = s, ℓn + m ≡ s ) , ℓ ... s − , ℓn + m ≡ s ) , ℓ ... , char K = 3;6 s, ℓn + m ≡ s ) , ℓ ... , char K = 3;0 , otherwise. (6) If r = 5 , then dim K Im δ t = s − , ℓn + m ≡ s ) , ℓ ... , char K = 3;6 s, ℓn + m ≡ s ) , ℓ ... , char K = 3;2 s, ℓn + m ≡ s ) , ℓ ... , otherwise. (7) If r = 6 , then dim K Im δ t = s − , ℓn + m ≡ s ) , ℓ ... s − , ℓn + m ≡ s ) , ℓ ... , char K = 2;5 s, ℓn + m ≡ s ) , ℓ ... , char K = 2;0 , otherwise. (8) If r = 7 , then dim K Im δ t = s, ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (9) If r = 8 , then dim K Im δ t = s − , ℓn + m ≡ s ) , ℓ ... s, ℓn + m ≡ s ) , ℓ ... , otherwise. (10) If r = 9 , then dim K Im δ t = s, ℓn + m ≡ s ) , ℓ ... s − , ℓn + m ≡ s ) , ℓ ... , otherwise. (11) If r = 10 , then dim K Im δ t = s, ℓn + m ≡ s ) , ℓ ... s − , ℓn + m ≡ s ) , ℓ ... , char K = 3;6 s, ℓn + m ≡ s ) , ℓ ... , char K = 3;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 17 (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 ∈ { , , , } , ℓn + m ≡ s ) , ℓ ... ; (2) r ∈ { , } , ℓn + m ≡ s ) , ℓ ... , char K = 3 ; (3) r ∈ { , } , ℓn + m ≡ s ) ; (4) r ∈ { , , } , ℓn + m ≡ s ) , ℓ ... ; (5) r = 3 , ℓn + m ≡ s ) , ℓ ... , char K = 2 ; (6) r = 3 , ℓn + m ≡ s ) , ℓ ... ; (7) r ∈ { , } , ℓn + m ≡ s ) , ℓ ... , char K = 3 ; (8) r = 4 , ℓn + m ≡ s ) , ℓ ... , char K = 2 ; (9) r = 5 , ℓn + m ≡ s ) , char K = 3 ; (10) r ∈ { , } , ℓn + m ≡ s ) , ℓ ... , char K = 2 .In other cases dim K HH t ( R ) = 0 .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 14 – 16. (cid:3) Theorem 18 (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 ) = 7 , if t = 0 . ( b ) dim K HH t ( R ) = 2 , if one of the following conditions takes place :(1) r ∈ { , } , t > , ℓ ... , char K = 3 ; (2) r ∈ { , } , ℓ ... , char K = 3 . ( c ) dim K HH t ( R ) = 1 , if one of the following conditions takes place :(1) r ∈ { , } , t > , ℓ ... , char K = 3 ; (2) r ∈ { , } ; (3) r ∈ { , } , ℓ ... ; (4) r ∈ { , } , ℓ ... , char K = 2 ; (5) r ∈ { , } , ℓ ... , char K = 3 ; (6) r = 5 , char K = 3 ; (7) r ∈ { , } , ℓ ... , char K = 2 ; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (8) r ∈ { , } , ℓ ... . ( 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 2. For s = 1 introduce theset of generators Y (1) t , Y (2) t , . . . Y (28) 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 2 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
10) and denote by κ = ( − ⌊ ℓ ⌋ .(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 = κe jn ⊗ e jn , i = j ;0 , otherwise.If s j < s , then y ij = e jn + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If 3 s j < s , then y ij = e jn + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If 5 s j < s , then y ij = κe jn +5 ⊗ e jn +5 , i = j ;0 , otherwise.(2) Y (2) t is an (6 s × s ) matrix with a single nonzero element: y s, s = w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 . (3) Y (3) t is an (7 s × s ) matrix with a single nonzero element: y s, s = κw jn +5 → ( j +1) n ⊗ e jn +5 . (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 jn → jn + g ( j + s ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then y ij = 0. If 5 s j < s , then y ij = κw jn + g ( j )+1 → jn +5 ⊗ e jn + g ( j )+1 , 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 s, s = w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 . (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 jn → jn + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then y ij = w jn → jn + g ( j )+1 ⊗ e jn , i = j − s ;0 , otherwise.If 2 s j < s , then y ij = 0.If 4 s j < s , then y ij = w jn + g ( j )+1 → ( j +1) n ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If 5 s j < s , then y ij = 0.If 6 s j < s , then y ij = w jn +5 → ( j +1) n + g ( j ) ⊗ e jn +5 , i = j − s ;0 , otherwise.If 7 s j < s , then y ij = 0.(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 jn → jn + g ( j + s )+1 ⊗ e jn , i = ( j ) s ;0 , otherwise.If 2 s j < s , then y ij = − κw jn + g ( j ) → jn +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If 4 s j < s , then y ij = − κw jn + g ( j )+1 → ( j +1) n ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If 6 s j < s , then y ij = 0. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (8) Y (8) t is an (9 s × s ) matrix with a single nonzero element: y s, s = w jn + g ( j )+1 → ( j +1) n + g ( j )+1 ⊗ e jn + g ( j )+1 . (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 = κe jn ⊗ e jn , i = j − s ;0 , otherwise.If 2 s j < s , then y ij = e jn + g ( j ) ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If 4 s j < s , then y ij = 0.If 6 s j < s , then y ij = e jn + g ( j )+1 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If 8 s j < s , then y ij = κe jn +5 ⊗ e jn +5 , i = j − s ;0 , otherwise.(10) Y (10) t is an (9 s × s ) matrix with a single nonzero element: y s, s = w jn +5 → ( j +1) n +5 ⊗ e jn +5 . (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 jn → jn + g ( j ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then y ij = 0.If 2 s j < s , then y ij = κw jn + g ( j ) → ( j +1) n ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If 3 s j < s , then y ij = 0.If 4 s j < s , then y ij = − κw jn + g ( j )+1 → jn +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If 5 s j < s , then y ij = 0. If 6 s j < s , then y ij = w jn +5 → ( j +1) n + g ( j )+1 ⊗ e jn +5 , i = j − s ;0 , otherwise.If 7 s j < s , then y ij = 0.(12) Y (12) t is an (8 s × s ) matrix, whose elements y ij have the following form:If 0 j < s , then y ij = − w jn → jn + g ( j + s ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then y ij = 0.If 4 s j < s , then y ij = κw jn + g ( j )+1 → jn +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If 5 s j < s , then y ij = 0.(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 = − f ( j, s ) e jn + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If 3 s j < s , then y ij = 0.If 5 s j < s , then y ij = − f ( j, s ) e jn + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 , otherwise.If 7 s j < s , then y ij = 0.If 8 s j < s , then y ij = − κw jn +5 → ( j +1) n ⊗ e jn +5 , i = j − s ;0 , otherwise.(14) Y (14) t is an (9 s × s ) matrix with a single nonzero element: y , = κw jn → ( j +1) n ⊗ e jn . (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 = e jn ⊗ e jn , i = j ;0 , otherwise.If s j < s , then y ij = 0. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If 3 s j < s , then y ij = w jn + g ( j + s ) → jn + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j − s ;0 , otherwise.If 5 s j < s , then y ij = 0.If 7 s j < s , then y ij = e jn +5 ⊗ e jn +5 , i = j − s ;0 , otherwise.If 8 s j < s , then y ij = 0.(16) Y (16) t is an (8 s × s ) matrix, whose elements y ij have the following form:If j = 0, then y ij = w jn → jn + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If 0 < j < s , then y ij = 0.If j = 2 s , then y ij = − κw jn + g ( j ) → jn +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If 2 s < j < s , then y ij = 0.(17) Y (17) t is an (8 s × s ) matrix, whose elements y ij have the following form:If j = 0, then y ij = w jn → jn + g ( j + s )+1 ⊗ e jn , i = j ;0 , otherwise.If 0 < j < s , then y ij = 0.If j = 2 s , then y ij = w jn + g ( j ) → jn +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If 2 s < j < s , then y ij = 0.(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 jn + g ( j + s ) → jn + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If 3 s j < s , then y ij = 0.If 5 s j < s , then y ij = − κw jn +5 → ( j +1) n ⊗ e jn +5 , i = j ;0 , otherwise. (19) Y (19) t is an (7 s × s ) matrix, whose elements y ij have the following form:If 0 j < s , then y ij = 0.If j = s , then y ij = κw jn + g ( j + s ) → ( j +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s < j < s , then y ij = 0.If j = 2 s , then y ij = κw jn + g ( j + s ) → ( j +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If 2 s < j < s , then y ij = 0.(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 = κw jn → jn +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then y ij = κw jn + g ( j + s ) → ( j +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If 2 s j < s , then y ij = 0.If 3 s j < s , then y ij = − w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If 4 s j < s , then y ij = 0.If 6 s j < s , then y ij = − w jn +5 → ( j +1) n + g ( j )+1 ⊗ e jn +5 , i = j − s ;0 , otherwise.(21) Y (21) t is an (6 s × s ) matrix with a single nonzero element: y , = − κw jn → ( j +1) n ⊗ e jn . (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 = 0.If s j < s , then y ij = f ( j, s ) e jn + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If 3 s j < s , then y ij = f ( j, s ) e jn + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If 5 s j < s , then y ij = 0.(23) Y (23) t is an (6 s × s ) matrix with a single nonzero element: y s, s = w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 . (24) Y (24) t is an (6 s × s ) matrix with a single nonzero element: y , = w jn → ( j +1) n ⊗ e jn . (25) Y (25) t is an (6 s × s ) matrix with a single nonzero element: y s,s = w jn + g ( j + s ) → ( j +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) . (26) Y (26) t is an (6 s × s ) matrix with a single nonzero element: y s, s = − w jn + g ( j + s ) → ( j +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) . (27) Y (27) t is an (6 s × s ) matrix with a single nonzero element: y s, s = − w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 . (28) Y (28) t is an (6 s × s ) matrix with a single nonzero element: y s, s = w jn +5 → ( j +1) n +5 ⊗ e jn +5 .
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
10) and denote by κ = ( − ⌊ ℓ ⌋ .(2) For translate Ω t represent t in the form t = 11 ℓ + r (0 r
10) and denote by κ = ( − ℓ .Define the helper functions z : Z × Z × Z → Z and z : Z × Z × Z → Z , which act in thefollowing way: z ( x, ℓ, k ) = ( x − ℓk ) s + ( ℓs ) s , z ( x, ℓ, k ) = ( x − ℓk ) s + (( ℓ + 1) s ) s Proposition 19 (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 = κe ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn +5 , 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 ) n + g ( j ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn +5 , 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 = κe ( j + m − n +5 ⊗ e jn , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn +5 , 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 ) n + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j ;0 , otherwise. (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 = κe ( j + m − n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s ) ⊗ e jn + g ( j ) , i = j ;0 , otherwise. If s j < s , then b ij = e ( j + m ) n + g ( j ) ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn +5 , 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 ) n + g ( j ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , 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 = κe ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn +5 , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn +5 , 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 : If j < s , then b ij = e ( j + m ) n + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j ) ⊗ e jn +5 , 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 = κe ( j + m − n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise. If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn +5 , 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 = κe ( j + m ) n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ e jn +5 , 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 = κe ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn +5 , 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 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E Proposition 20 (Translates for the case 2) . (I) Let r ∈ N , r < . 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 s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 . (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 s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j ) . (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 s + z ( − ,ℓ ,n ) ,s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) . (4) If r = 3 , then Ω ( Y (2) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z ( − ,ℓ ,n ) ,z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn . (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 s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 . (6) If r = 5 , then Ω ( Y (2) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m +1) n + g ( j )+1 → ( j + m +2) n + g ( j )+1 ⊗ e jn +5 . (7) If r = 6 , then Ω ( Y (2) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + z ( − , ℓ , n ) , then b ij = 0 .If j = 3 s + z ( − , ℓ , n ) , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s + z ( − , ℓ , n ) < j < s + z ( − , ℓ , n ) , then b ij = 0 .If j = 5 s + z ( − , ℓ , n ) , then b ij = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s + z ( − , ℓ , n ) < j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (2) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z ( − ,ℓ ,n ) ,z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn . (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 s + z ( − ,ℓ ,n ) ,s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) . (10) If r = 9 , then Ω ( Y (2) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n + g ( j + s )+1 ⊗ e jn +5 . (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 s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 . (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 , and coefficients multiplied by ( − ℓ . Proposition 21 (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 one nonzero element thatis of the following form : b s +( − ℓ ) s , s +( − ℓ ) s = κw ( j + m ) n +5 → ( j + m +1) n ⊗ e jn +5 . (2) If r = 1 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s + ( − ℓ ) s , then b ij = 0 .If j = s + ( − ℓ ) s , then b ij = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j + s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s + ( − ℓ ) s < j < s + ( − ℓ ) s , then b ij = 0 .If j = 2 s + ( − ℓ ) s , then b ij = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j + s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s + ( − ℓ ) s < j < s + ( − ℓ ) s , then b ij = 0 .If j = 3 s + ( − ℓ ) s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s + ( − ℓ ) s < j < s + ( − ℓ ) s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j = 4 s + ( − ℓ ) s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s + ( − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 5 s + ( − − ℓ ) s , then b ij = κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 6 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (3) If r = 2 , then Ω ( Y (3) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < ( − ℓ ) s , then b ij = 0 .If j = ( − ℓ ) s , then b ij = w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If ( − ℓ ) s < j < s + ( − ℓ ) s , then b ij = 0 .If j = s + ( − ℓ ) s , then b ij = w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j − s ;0 , otherwise.If s + ( − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 6 s + ( − − ℓ ) s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 7 s + ( − − ℓ ) s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (4) If r = 3 , then Ω ( Y (3) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + ( − − ℓ ) s , then b ij = 0 .If j = 6 s + ( − − ℓ ) s , then b ij = − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 . If j = 7 s + ( − − ℓ ) s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 8 s + ( − − ℓ ) s , then b ij = − κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ; κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (3) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ; κe ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = − κe ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 4 s + ( − − ℓ ) s , then b ij = − κe ( j + m ) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; κe ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 4 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 5 s + ( − − ℓ ) s , then b ij = − κe ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (3) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + ( − − ℓ ) s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j = s + ( − − ℓ ) s , then b ij = − e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = − e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j − s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 4 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j − s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 5 s + ( − − ℓ ) s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 6 s + ( − − ℓ ) s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (7) If r = 6 , then Ω ( Y (3) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = w ( j + m ) n → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j ; − w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 . If j = s + ( − − ℓ ) s , then b ij = w ( j + m ) n → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j − s ; − w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s + ( − − ℓ ) s , then b ij = 0 .If j = s + ( − − ℓ ) s , then b ij = − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 5 s + ( − − ℓ ) s , then b ij = − κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn +5 , i = j + s ; − κw ( j + m +1) n + g ( j ) → ( j + m +2) n ⊗ e jn +5 , i = j + 2 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (9) If r = 8 , then Ω ( Y (3) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = κe ( j + m ) n +5 ⊗ w jn → ( j +1) n , i = ( j + 1) s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = s + ( − − ℓ ) s , then b ij = κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ; − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 2 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = − κe ( j + m +1) n ⊗ w jn + g ( j + s ) → jn +5 , i = j + 3 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 4 s + ( − − ℓ ) s , then b ij = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (10) If r = 9 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = κe ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (3) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 . If j = ( − − ℓ ) s , then b ij = κw ( j + m ) n → ( j + m +1) n ⊗ e jn , i = j ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = s + ( − − ℓ ) s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 4 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 5 s + ( − − ℓ ) s , then b ij = − κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ; κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 4 s ; κe ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 5 s ;0 , otherwise.If s + ( − − ℓ ) 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 (3) t ) is a Ω r ( Y (3) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E Proposition 22 (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 ) n → ( j + m ) n + g ( j + s ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , 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 = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn , i = j + s ; κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 , i = j + s ; w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n → ( j + m +2) n ⊗ e jn +5 , i = j + s ;0 , otherwise. (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 = − e ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 . If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( 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 ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − e ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 3 s ;0 , otherwise.If s j < s , then b ij = 0 . (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 = 0 .If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j )+1 , i = j ; e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (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 = − w ( j + m − n +5 → ( j + m ) n + g ( j + s ) ⊗ e jn , i = j ; e ( j + m ) n + g ( j + s ) ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j ; κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn + g ( 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 ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (4) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ; w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 , otherwise. If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j ;0 , otherwise.If s j < s , then b ij = 0 . (7) If r = 6 , 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 ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j ) , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − e ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 , i = j + s ; − e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise. (9) If r = 8 , then Ω ( Y (4) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κw ( j + m − n +5 → ( j + m ) n +5 ⊗ e jn , i = j ; κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − e ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ;0 , otherwise.If s j < s , then b ij = 0 . (10) If r = 9 , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 . If s j < s , then b ij = − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (4) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κe ( j + m +1) n ⊗ w jn → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , 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) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 4 s ; κe ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 5 s ;0 , otherwise. (II) 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 23 (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 : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j < s , then b ij = w ( j + m ) n → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( 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 ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (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 = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn , i = j − s ; w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( 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) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise. If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn +5 , 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 − n +5 → ( j + m ) n + g ( j ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( 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) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 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 = w ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn , i = j ; w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , 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) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; e ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 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 − n +5 → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j ; w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ; e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 7 s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = j + s ; e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , 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 = w ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn , i = j ; w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn , i = j + s ; e ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; e ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 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 = w ( j + m ) n → ( j + m ) n +5 ⊗ e jn , i = j ; w ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + s ; w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n ⊗ w jn + g ( j + s ) → ( j +1) n , 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) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n , 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) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 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 : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j < s , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn , i = j ; w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ; e ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; e ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 . (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 = w ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + s ; w ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ; w ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 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 = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn , i = j − s ;0 , otherwise. If s j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m +1) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n +5 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( 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 = w ( j + m ) n → ( j + m ) n +5 ⊗ e jn , i = j ; w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + s ; w ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ; w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ; w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ; e ( j + m ) n +5 ⊗ w jn → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 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 24 (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 : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j < s , then b ij = w ( j + m ) n → ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = ( j ) s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( j )+1 , 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 = κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn , i = j ; κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn , i = j + s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn , i = j ; κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j )+1 , i = j − s ; f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j )+1 → jn +5 , i = j − s ( j − , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , 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 j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s (8 − j );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 = κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ( j − f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + 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 = f ( j, s ) w ( j + m − n +5 → ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = j − sj ; − f ( j, s ) w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j ) , i = j + 2 s ; f ( j, s ) e ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 6 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , 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 j < s , then b ij = 0 .If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n ,i = ( j + 1) s + s (2 − j ); − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 3 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 : If j < s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + s ; − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ( j − , otherwise.If s j < s , then b ij = f ( j, s ) w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + s (7 − j );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 j < s , then b ij = 0 .If s j < s , then b ij = f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ; f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ; − f ( j, s ) e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , 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 = κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ; − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ( j − , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; − κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 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 : If j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn , i = j − s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j ) → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j ) , i = j − s ; − e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j ) → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m +1) n +5 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ;0 , otherwise. (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 = − κw ( j + m ) n → ( j + m ) n +5 ⊗ e jn , i = j ; κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + s ; κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ; − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , 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) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 4 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 25 (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 s + z (0 ,ℓ , , s + z (0 ,ℓ , = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j )+1 . (2) If r = 1 , then Ω ( Y (8) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b (6 s + z ( − ,ℓ , s +2 s +( ℓ s ) s , s + z ( − ,ℓ , = − κ w ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) . (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 (3 s + z ( − ,ℓ , s + s +( ℓ s ) s , s + z ( − ,ℓ , = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) . (4) If r = 3 , then Ω ( Y (8) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b (2 s + z ( − ,ℓ , s +2 s +( ℓ s ) s , s + z ( − ,ℓ , = κκ e ( j + m +1) n +5 ⊗ w jn + g ( j ) → ( j +1) n + g ( j ) . (5) If r = 4 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( s + z ( − ,ℓ , s +2 s +( ℓ s ) s ,s + z ( − ,ℓ , = − w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) . (6) If r = 5 , then Ω ( Y (8) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b ( s + z ( − ,ℓ , s +2 s +( ℓ s ) s ,s + z ( − ,ℓ , = κκ e ( j + m +2) n ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) . (7) If r = 6 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( s + z ( − ,ℓ , s + s +( ℓ s ) s ,s + z ( − ,ℓ , = e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (8) If r = 7 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( s + z ( − ,ℓ , s +2 s +( ℓ s ) s ,s + z ( − ,ℓ , = − w ( j + m +1) n +5 → ( j + m +2) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) . (9) If r = 8 , then Ω ( Y (8) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b (2 s + z ( − ,ℓ , s + s +( ℓ s ) s , s + z ( − ,ℓ , = − e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j ) → ( j +1) n + g ( j ) . (10) If r = 9 , then Ω ( Y (8) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z ( − ,ℓ , ,s + z ( − ,ℓ , = − w ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 . (11) If r = 10 , then Ω ( Y (8) t ) is described with (8 s × s ) -matrix with one nonzero elementthat is of the following form : b ( s + z ( − ,ℓ , s +3 s,z ( − ,ℓ , = κ w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 . (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 26 (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 j < s , then b ij = 0 .If s j < s , then b ij = κe ( j + m ) n ⊗ e jn , i = j − s ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j ) ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn +5 , 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 ) n + g ( j + s ) ⊗ e jn , i = ( j ) s + f ( j, s ) s ;0 , otherwise.If s j < s , then b ij = − κf ( j, s ) w ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( j ) , i = j ; κf ( j, s ) w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 2 s ; κf ( j, s ) e ( j + m +1) n ⊗ w jn + g ( j ) → jn +5 , i = ( j ) s + 6 s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) e ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n ,i = ( j + 1) s + f ( j, s ) s ; f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + ( f ( j, s ) + 2) s ; − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn +5 , i = ( j ) s + 6 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 = − κw ( j + m − n +5 → ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = − e ( j + m ) n + g ( j ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − κe ( j + m ) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κe ( j + m +1) n ⊗ e jn +5 , 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 = − f ( j, s ) e ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = ( j ) s + f ( j, s ) s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) e ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) e ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j + s ) ⊗ e jn +5 , i = ( j ) s + ( f ( j, s ) + 6) 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 = κe ( j + m − n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j + s ; − e ( j + m ) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise. If s j < s , then b ij = − κe ( j + m +1) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ; − κw ( j + m ) n +5 → ( j + m +1) n ⊗ e jn +5 , 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 = − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn , i = j + s ; − κe ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; − κe ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) e ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j + s ; κf ( j, s ) w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s + (1 − f ( j, s )) s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + f ( j, s ) s ; − f ( j, s ) e ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , i = ( j ) s + (6 + f ( j, s )) s ;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 = κe ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ e jn +5 , 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 = − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn , i = j + s ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ; − κe ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; − κe ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ; w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ; − e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s + (1 − f ( j, s )) s ; − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j + s ; w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ; − κe ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; − κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ e jn +5 , i = j + s ; − κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ e jn +5 , 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 ) n + g ( j ) ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m − n +5 → ( j + m ) n + g ( j ) ⊗ e jn , i = j − s ; e ( j + m ) n + g ( j ) ⊗ w jn → jn + g ( j + s )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = − e ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κe ( j + m +1) n ⊗ e jn +5 , 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 = − κe ( j + m ) n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j ; − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − e ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n ⊗ e jn +5 , 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 ) n → ( j + m ) n + g ( j )+1 ⊗ e jn , i = ( j ) s ; f ( j, s ) w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = ( j ) s + (1 + f ( j, s )) s ; f ( j, s ) e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = ( j ) s + (3 + f ( j, s )) s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ; κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ; κf ( j, s ) e ( j + m ) n +5 ⊗ w jn + g ( j ) → jn +5 , i = ( j ) s + 5 s ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j − s ;0 , otherwise. If s j < s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 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 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 ) n → ( j + m ) n + g ( j ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( 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 ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (2) If r = 1 , then Ω ( Y (11) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ; w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (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 = − w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j ; − e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ;0 , otherwise. If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (4) If r = 3 , then Ω ( Y (11) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j + s ; w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , 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) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn +5 , i = j + s ;0 , otherwise. (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 = κw ( j + m − n +5 → ( j + m ) n +5 ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ; − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 7 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j + s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s + s ; − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ; − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 7 s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 . (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 = 0 .If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j + s ; e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j + s ;0 , otherwise. (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 = − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 4 s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , 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) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 4 s ; − κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn +5 , i = j + 3 s ;0 , otherwise. (8) If r = 7 , then Ω ( Y (11) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn → jn + g ( j ) , i = j + 2 s ; − w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn → jn + g ( j + s )+1 , i = j + 3 s ; − w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j ) , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn + g ( j )+1 , i = j ; − κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κw ( j + m +1) n + g ( j ) → ( j + m +2) n ⊗ e jn +5 , i = j ; − κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn +5 , i = j + s ;0 , otherwise. (9) If r = 8 , then Ω ( Y (11) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κe ( j + m ) n +5 ⊗ w jn → ( j +1) n , i = ( j + 1) s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; − κw ( j + m +1) n + g ( j ) → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 s ;0 , otherwise. (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 ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − κe ( j + m +1) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n +5 → ( j + m +2) n ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn + g ( 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) n +5 → ( j + m +2) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = − w ( j + m +2) n → ( j + m +2) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise. (11) If r = 10 , 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 If j < s , then b ij = κw ( j + m ) n → ( j + m ) n +5 ⊗ e jn , i = j ; κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + s ; − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ; κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ; − κe ( j + m ) n +5 ⊗ w jn → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = − κe ( j + m +1) n ⊗ w jn → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j )+1 , i = j − s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) 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 (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 elements b ij : If j < s , then b ij = − w ( j + m ) n → ( j + m ) n + g ( j + s ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (2) If r = 1 , then Ω ( Y (12) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn , i = j + s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j − s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + s ; − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = 0 . (3) If r = 2 , then Ω ( Y (12) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m − n +5 → ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = j − s ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 . (4) If r = 3 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn , i = j + s ; κe ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (12) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κe ( j + m ) n +5 ⊗ w jn → ( j +1) n , i = ( j + 1) s ; − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ; κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 7 s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j + s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 3 s ; − e ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 7 s ;0 , otherwise. (6) If r = 5 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κe ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ; κe ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 . (7) If r = 6 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κw ( j + m ) n → ( j + m +1) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + j s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) ,i = ( j + 1) s + ( j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − κw ( j + m +1) n → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn +5 , i = j + 2 s ;0 , otherwise. (8) If r = 7 , then Ω ( Y (12) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn + g ( j )+1 , i = j ; − κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 . (9) If r = 8 , then Ω ( Y (12) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ; − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise. (10) If r = 9 , then Ω ( Y (12) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = − κe ( j + m +2) n ⊗ w jn + g ( j )+1 → ( j +1) n + g ( j ) , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m +1) n + g ( j )+1 → ( j + m +2) n + g ( j ) ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (12) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 . If s j < s , then b ij = e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j ) → ( j +1) n + g ( j ) , i = ( j + 1) s + s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j ) , i = j − s ; − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n + g ( j ) , i = ( j + 1) s + s ; − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 3 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 , i = j − 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 (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 j < s , then b ij = 0 .If s j < s , then b ij = − f ( j, s ) e ( j + m ) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − f ( j, s ) e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n +5 → ( j + m +1) n ⊗ e jn +5 , 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 = f ( j, s ) e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = − κf ( j, s ) w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = − κf ( j, s ) w ( j + m ) n +5 → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = f ( j, s ) e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + ( j − s ; f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = ( j ) s + 6 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 j < s , then b ij = 0 .If s j < s , then b ij = − f ( j, s ) e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m ) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn +5 → ( j +1) n , 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 = − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn , i = j ; κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn , i = j + s ;0 , otherwise.If s j < s , then b ij = − κf ( j, s ) w ( j + m ) n +5 → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ; − f ( j, s ) e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2(1 + f ( j, s )) s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + ( j − 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 = − κw ( j + m − n +5 → ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m ) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j + 3 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 = κe ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ; κe ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + f ( j, s ) s ; − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j + s ; e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2(1 + f ( j, s )) s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j + s ; κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j + 2 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 = − f ( j, s ) e ( j + m ) n + g ( j + s ) ⊗ w jn → jn + g ( j + s ) , i = j + 2 f ( j, s ) s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κe ( j + m +1) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ e jn +5 , i = j + 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 = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn , i = j + s ;0 , otherwise. If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + f ( j, s ) s ; f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 , i = j + s ; f ( j, s ) e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n + g ( j )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ; − κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ; − κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 5 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 = − f ( j, s ) w ( j + m − n +5 → ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = ( j ) s ; − e ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j + s ) , i = ( j ) s + (1 + f ( j, s )) s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ; − κe ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ; − κe ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = f ( j, s ) e ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + ( j − 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 j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m ) n +5 → ( j + m +1) n ⊗ e jn , i = j − s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j )+1 → jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = − κe ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , 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 ) n → ( j + m ) n + g ( j + s ) ⊗ e jn , i = ( j ) s ; − f ( j, s ) e ( j + m ) n + g ( j + s ) ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ; − κf ( j, s ) w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j ) → jn +5 , i = ( j ) s + 5 s ;0 , otherwise. If s j < s , then b ij = κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ; κf ( j, s ) e ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = ( j ) s + 5 s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + ( j − s ; e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + ( j − 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 ( − ℓ ) s , ( − ℓ ) s = κw ( j + m ) n → ( j + m +1) n ⊗ e jn . (2) If r = 1 , then Ω ( Y (14) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b ( − ℓ ) s , s +( − − ℓ ) s = κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ w jn + g ( j )+1 → ( j +1) n . (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 s +( − − ℓ ) s , s +( − − ℓ ) s = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 . (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 s +( − − ℓ ) s , s +( − − ℓ ) s = − w ( j + m +2) n → ( j + m +2) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+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 +( − − ℓ ) s , s +( − − ℓ ) s = − e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+1 . (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 s +( − − ℓ ) s , s +( − − ℓ ) s = w ( j + m +1) n +5 → ( j + m +2) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+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 s +( − − ℓ ) s , s +( − − ℓ ) s = − κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n + g ( j )+1 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (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 s +( − − ℓ ) s , s +( − − ℓ ) s = − w ( j + m +2) n → ( j + m +2) n + g ( j + s ) ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s )+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 s +( − − ℓ ) s , s +( − − ℓ ) s = κw ( j + m +1) n → ( j + m +2) n ⊗ w jn + g ( j )+1 → jn +5 . (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 s +( − − ℓ ) s ,s +( − − ℓ ) s = − κw ( j + m +1) n → ( j + m +2) n ⊗ w jn → jn + g ( j + s ) . (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 (2 s +( − − ℓ ) s +1) s , s +( − − ℓ ) s = κw ( j + m +1) n → ( j + m +2) n ⊗ w jn + g ( j ) → ( j +1) n . (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 = e ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = e ( j + m ) n +5 ⊗ e jn +5 , 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 ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = ( j ) s + f ( j, s ) s ;0 , otherwise. If s j < s , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ; e ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( 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 +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn +5 , 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 = e ( j + m − n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+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 +1) n ⊗ e jn +5 , i = j ;0 , otherwise. (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 = w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn , i = j ; e ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , i = j + 2 f ( j, s ) 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 = e ( j + m ) n ⊗ e jn , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n +5 ⊗ e jn +5 , 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 = w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn , i = j + s ; e ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ; w ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + ( j − s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s + ( j − s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise. If s j < s , then b ij = w ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; e ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 5 s ; w ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j + s ;0 , otherwise. (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 ) n + g ( j ) ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n → ( j + m ) n + g ( j ) ⊗ e jn , i = j − s ; e ( j + m ) n + g ( j ) ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j + s ; e ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; e ( j + m +1) n ⊗ e jn +5 , i = j + 2 s ;0 , otherwise. (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 = w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn , i = j + s ; e ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + ( j − s ; w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = e ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ; w ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn +5 , i = j + s ;0 , otherwise. (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 ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ; w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j − s ; e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ; w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ; e ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise. If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn +5 , 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 = e ( j + m ) n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ;0 , otherwise. (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 ) n → ( j + m ) n + g ( j ) ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = e ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ; e ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise. (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 elements b ij : If j < ( − ℓ ) s , then b ij = 0 .If j = ( − ℓ ) s , then b ij = w ( j + m ) n → ( j + m ) n + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If ( − ℓ ) s < j < s + ( − ℓ ) s , then b ij = 0 . If j = 2 s + ( − ℓ ) s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s + ( − ℓ ) s < j < s , then b ij = 0 . (2) If r = 1 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < ( − ℓ ) s , then b ij = 0 .If j = ( − ℓ ) s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn , i = j ;0 , otherwise.If ( − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 5 s + ( − − ℓ ) s , then b ij = − κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m +1) n → ( j + m +2) n ⊗ e jn +5 , i = j + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (3) If r = 2 , then Ω ( Y (16) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s + ( − − ℓ ) s , then b ij = 0 .If j = s + ( − − ℓ ) s , then b ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 2 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 4 s + ( − − ℓ ) s , then b ij = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s + ( − − ℓ ) s < j < s , then b ij = 0 . (4) If r = 3 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b ( − − ℓ ) s , ( − − ℓ ) s = κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn . (5) If r = 4 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = − κe ( j + m +1) n ⊗ w jn → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 2 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 4 s + ( − − ℓ ) s , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (16) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = e ( j + m +1) n + g ( j + s ) ⊗ w jn → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (7) If r = 6 , then Ω ( Y (16) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 . If j = 2 s + ( − − ℓ ) s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (16) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s + ( − − ℓ ) s , then b ij = 0 .If j = s + ( − − ℓ ) s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = − κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 6 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n + g ( j + s ) → ( j + m +2) n + g ( j + s ) ⊗ e jn +5 , i = j + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (9) If r = 8 , then Ω ( Y (16) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = κe ( j + m ) n +5 ⊗ w jn → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 3 s + ( − − ℓ ) s , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 4 s + ( − − ℓ ) s , then b ij = − w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (10) If r = 9 , then Ω ( Y (16) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < ( − − ℓ ) s , then b ij = 0 .If j = ( − − ℓ ) s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn , i = j ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 6 s + ( − − ℓ ) s , then b ij = − w ( j + m +1) n +5 → ( j + m +2) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s + ( − − ℓ ) s < j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (16) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + ( − − ℓ ) s , then b ij = 0 .If j = s + ( − − ℓ ) s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; − e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then b ij = 0 .If j = 5 s + ( − − ℓ ) s , then b ij = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j ;0 , otherwise.If s + ( − − ℓ ) 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 (16) t ) is a Ω r ( Y (16) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 33 (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 j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 .
00 MARIYA PUSTOVYKH If s j < s , then b ij = − κw ( j + m ) n +5 → ( j + m +1) n ⊗ e jn +5 , 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 : If j < s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j + s ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + ( j − s ; − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn +5 , i = j + (6 − 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 = κw ( j + m − n +5 → ( j + m ) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 , 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 : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn , i = j ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s + (4 − j ) s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − κe ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise. (5) If r = 4 , then Ω ( Y (18) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m − n +5 → ( j + m ) n + g ( j + s ) ⊗ e jn , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j ) , i = j ; − e ( j + m ) n + g ( j + s )+1 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = κe ( j + m ) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (18) t ) is described with (6 s × s ) -matrix with the following elements b ij :
02 MARIYA PUSTOVYKH If j < s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn , i = j + s ; κe ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; κe ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + (2 − j ) s ; − w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 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 = f ( j, s ) w ( j + m ) n → ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = ( j ) s ; − f ( j, s ) w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j + s ) , i = ( j ) s + (2 − j ) s ;0 , otherwise.If s j < s , then b ij = − κf ( j, s ) w ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = κe ( j + m +1) n ⊗ w jn + g ( j )+1 → ( j +1) n , 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) n → ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − e ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − e ( j + m +1) n + g ( j + s ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise. (8) If r = 7 , then Ω ( Y (18) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn , i = j ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + s ; κe ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 3 s ; κe ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j ) , i = j ; w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 2 s ; − e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j ) → jn +5 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = 0 . (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 − n +5 → ( j + m ) n + g ( j + s ) ⊗ e jn , i = j ; e ( j + m ) n + g ( j + s ) ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = e ( j + m ) n + g ( j + s ) ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.
04 MARIYA PUSTOVYKH If s j < s , then b ij = 0 .If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + (8 − j ) 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 = κw ( j + m ) n +5 → ( j + m +1) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j − s ; − e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κe ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , 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 = f ( j, s ) w ( j + m ) n → ( j + m ) n + g ( j + s )+1 ⊗ e jn , i = ( j ) s ; − w ( j + m ) n + g ( j + s ) → ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j ) , i = j + s ; e ( j + m ) n + g ( j + s )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ; − κf ( j, s ) w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ; − κe ( j + m ) n +5 ⊗ w jn + g ( j ) → jn +5 , i = ( j ) s + 5 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 (18) t ) is a Ω r ( Y (18) t ) , whose left components twisted by σ ℓ . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E Proposition 34 (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 = κw ( j + m ) n → ( j + m ) n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( 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 ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , 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 : If j < s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn +5 , i = j + s ;0 , otherwise.
06 MARIYA PUSTOVYKH (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 = κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 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 ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn → jn + g ( j ) , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ;0 , 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 = κw ( j + m − n +5 → ( j + m ) n +5 ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ; κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 7 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , 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 j < s , then b ij = w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ⊗ w jn → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ; κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m +2) n → ( j + m +2) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 3 s ;0 , 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 = 0 .If s j < s , then b ij = − κw ( j + m ) n → ( j + m +1) n ⊗ e jn , i = j − s ; κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j ) → ( j +1) n , i = ( 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 :
08 MARIYA PUSTOVYKH If j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn → jn + g ( j ) , i = j + 2 s ; w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j )+1 → ( j + m +2) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , 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 = − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + s ; − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (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 ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = κw ( j + m +1) n → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = w ( j + m +1) n +5 → ( j + m +2) n + g ( j ) ⊗ w jn +5 → ( j +1) n , i = ( 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 = κw ( j + m ) n → ( j + m ) n +5 ⊗ e jn , i = j ;0 , otherwise.If s j < s , then b ij = 0 .If s j < s , then b ij = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ; w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j ) → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; κw ( j + m +1) n + g ( j )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 4 s ; κw ( j + m +1) n +5 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 5 s ;0 , otherwise. (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 35 (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 = 0 .
10 MARIYA PUSTOVYKH If s j < s , then b ij = f ( j, s ) e ( j + m ) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s j < s , then b ij = f ( j, s ) e ( j + m ) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = 0 . (2) If r = 1 , then Ω ( Y (22) t ) is described with (6 s × s ) -matrix with the following elements b ij : If j < s , then b ij = κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn , i = j ; − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn , i = j + s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ; κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + ( j − s ; − f ( j, s ) w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ; f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ; f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , i = ( j ) s + 6 s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n ,i = ( j + 1) s + ( j − s ; f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + ( j − s ; − f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j + s ; − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = ( j ) s + 6 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ; − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 3 s ;0 , otherwise. (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 = − w ( j + m − n +5 → ( j + m ) n + g ( j ) ⊗ e jn , i = ( j ) s ; − f ( j, s ) e ( j + m ) n + g ( j ) ⊗ w jn → jn + g ( j + s )+1 , i = ( j ) s + (4 − j ) s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m ) n + g ( j )+1 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j − s ;0 , otherwise.If s j < s , then b ij = 0 . (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 = − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn , i = j ; κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ e jn , i = j + s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + ( j − s ; − f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j + s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j + s ; − f ( j, s ) e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = ( j ) s + (10 − j ) s ;0 , otherwise.
12 MARIYA PUSTOVYKH If s j < s , then b ij = − κw ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn +5 , i = j + s ; − κw ( j + m +1) n + g ( j ) → ( j + m +2) n ⊗ e jn +5 , i = j + 2 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 : If j < s , then b ij = w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ e jn , i = ( j ) s ; − w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = ( j ) s + (3 − j ) s ; − e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = ( j ) s + (7 − j ) s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ; κf ( j, s ) w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − κf ( j, s ) w ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ; − κf ( j, s ) w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 3 s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j + s ;0 , 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 = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = κe ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + s ; κe ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + 2 s ; − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 3 s ; − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s + (3 − j ) s ; − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j ) , i = j ; f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 , i = j − s ; e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → jn +5 , i = ( j ) s + (13 − j ) s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ; κw ( j + m +1) n + g ( j + s )+1 → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ;0 , otherwise. (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 = f ( j, s ) e ( j + m ) n + g ( j ) ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.
14 MARIYA PUSTOVYKH If s j < s , then b ij = − κe ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ; − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j ) → jn +5 , i = j + 5 s ; κe ( j + m +1) n ⊗ w jn + g ( j ) → jn +5 , i = j + 6 s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn + g ( j ) , i = j − s ; κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + 3 s ; − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j ) → jn +5 , i = j + 4 s ; − κe ( j + m +1) n ⊗ w jn + g ( j ) → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) w ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = − e ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 5 s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , i = j + s ; − w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , i = j + 2 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 3 s ; e ( j + m +1) n + g ( j )+1 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 6 s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 , 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 : If j < s , then b ij = − κe ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; − κe ( j + m +1) n ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ; − w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 3 s ; − e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) , i = j + s ; e ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → jn +5 , i = j + 5 s ;0 , otherwise.If s j < s , then b ij = e ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ; w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn + g ( 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 +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s + ( j − s ; − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j − s ; − w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m +1) n + g ( j )+1 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s + s ; − κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ e jn +5 , i = j − s ;0 , otherwise.If s j < s , then b ij = κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 4 s ; − κe ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 5 s ; κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ e jn +5 , i = j − s ;0 , otherwise. (9) If r = 8 , then Ω ( Y (22) t ) is described with (8 s × s ) -matrix with the following elements b ij :
16 MARIYA PUSTOVYKH If j < s , then b ij = e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ; w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − e ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j − s ; − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn + g ( j ) → jn + g ( j )+1 , i = j + s ;0 , otherwise.If s j < s , then b ij = κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ; κe ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ;0 , otherwise.If s j < s , then b ij = − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j − s ; − κe ( j + m +1) n ⊗ w jn + g ( j )+1 → jn +5 , i = j ;0 , otherwise.If s j < s , then b ij = − e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ; w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j − s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s j < s , then b ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; e ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 3 s ; − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ e jn +5 , i = j − s ;0 , otherwise. (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 = 0 .If s j < s , then b ij = − f ( j, s ) w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) , i = j ; − f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = − f ( j, s ) e ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s j < s , then b ij = κw ( j + m +1) n +5 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ;0 , otherwise. (11) If r = 10 , then Ω ( Y (22) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j ) , i = j + s ; − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ; − κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → jn + g ( j + s )+1 , i = j + 4 s ;0 , otherwise.If s j < s , then b ij = κf ( j, s ) w ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ; κf ( j, s ) w ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 2 s ; κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn +5 , i = ( j ) s + 5 s ;0 , otherwise.
18 MARIYA PUSTOVYKH If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; f ( j, s ) e ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + ( j − s ; − f ( j, s ) w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 , i = j ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = ( j ) s + 5 s ;0 , otherwise.If s j < s , then b ij = − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − f ( j, s ) w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j ) ,i = ( j + 1) s + (7 − j ) s ; − f ( j, s ) e ( j + m +1) n + g ( j + s )+1 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + (9 − j ) s ;0 , otherwise. (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 36 (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 s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 . (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 s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j ) . (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 s + z ( − ,ℓ ,n ) ,s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) . (4) If r = 3 , then Ω ( Y (23) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z ( − ,ℓ ,n ) ,z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn . (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 s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (6) If r = 5 , then Ω ( Y (23) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m +1) n + g ( j )+1 → ( j + m +2) n + g ( j )+1 ⊗ e jn +5 . (7) If r = 6 , then Ω ( Y (23) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + z ( − , ℓ , n ) , then b ij = 0 .If j = 3 s + z ( − , ℓ , n ) , then b ij = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s + z ( − , ℓ , n ) < j < s + z ( − , ℓ , n ) , then b ij = 0 .If j = 5 s + z ( − , ℓ , n ) , then b ij = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s + z ( − , ℓ , n ) < j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (23) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z ( − ,ℓ ,n ) ,z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn . (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 s + z ( − ,ℓ ,n ) ,s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) . (10) If r = 9 , then Ω ( Y (23) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m +1) n + g ( j + s )+1 → ( j + m +2) n + g ( j + s )+1 ⊗ e jn +5 . (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 s + z ( − ,ℓ ,n ) , s + z ( − ,ℓ ,n ) = w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 . (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 , and coefficients multiplied by ( − ℓ . Proposition 37 (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 , = w ( j + m ) n → ( j + m +1) n ⊗ e jn . (2) If r = 1 , then Ω ( Y (24) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b s, s = w ( j + m ) n → ( j + m +1) n ⊗ e jn +5 .
20 MARIYA PUSTOVYKH (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 s, s = w ( j + m ) n → ( j + m +1) n ⊗ e jn +5 . (4) If r = 3 , then Ω ( Y (24) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If j = 4 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .If j = 5 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (24) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b s,s = w ( j + m ) n → ( j + m +1) n ⊗ e jn . (6) If r = 5 , then Ω ( Y (24) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If j = 2 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .If j = 3 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (7) If r = 6 , then Ω ( Y (24) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j = 0 , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn , i = j ;0 , otherwise.If < j < s , then b ij = 0 .If j = 8 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn +5 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (24) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j = 4 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .If j = 5 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (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 s, s = w ( j + m ) n → ( j + m +1) n ⊗ e jn +5 . (10) If r = 9 , then Ω ( Y (24) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If j = s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .If j = 2 s , then b ij = w ( j + m ) n → ( j + m +1) n ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (11) If r = 10 , then Ω ( Y (24) t ) is described with (6 s × s ) -matrix with one nonzero elementthat is of the following form : b , = w ( j + m ) n → ( j + m +1) n ⊗ e jn . (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 , and coefficients multiplied by ( − ℓ . Proposition 38 (Translates for the case 25) . (I) Let r ∈ N , r < . r -translates of theelements Y (25) t are described by the following way. (1) If r = 0 , then Ω ( Y (25) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) . (2) If r = 1 , then Ω ( Y (25) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b z (0 ,ℓ ,n ) ,z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn .
22 MARIYA PUSTOVYKH (3) If r = 2 , then Ω ( Y (25) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 . (4) If r = 3 , then Ω ( Y (25) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn +5 . (5) If r = 4 , then Ω ( Y (25) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + z (0 , ℓ , n ) , then b ij = 0 .If j = 2 s + z (0 , ℓ , n ) , then b ij = w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s + z (0 , ℓ , n ) < j < s + z (0 , ℓ , n ) , then b ij = 0 .If j = 4 s + z (0 , ℓ , n ) , then b ij = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s + z (0 , ℓ , n ) < j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (25) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z (0 ,ℓ ,n ) ,z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn . (7) If r = 6 , then Ω ( Y (25) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) . (8) If r = 7 , then Ω ( Y (25) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn +5 . (9) If r = 8 , then Ω ( Y (25) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 . (10) If r = 9 , then Ω ( Y (25) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 . (11) If r = 10 , then Ω ( Y (25) t ) is described with (6 s × s ) -matrix with one nonzero elementthat is of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (25) t ) is a Ω r ( Y (25) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 39 (Translates for the case 26) . (I) Let r ∈ N , r < . r -translates of theelements Y (26) t are described by the following way. (1) If r = 0 , then Ω ( Y (26) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) . (2) If r = 1 , then Ω ( Y (26) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b z (0 ,ℓ ,n ) ,z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn . (3) If r = 2 , then Ω ( Y (26) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 . (4) If r = 3 , then Ω ( Y (26) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn +5 . (5) If r = 4 , then Ω ( Y (26) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + z (0 , ℓ , n ) , then b ij = 0 .If j = 2 s + z (0 , ℓ , n ) , then b ij = − w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s + z (0 , ℓ , n ) < j < s + z (0 , ℓ , n ) , then b ij = 0 .If j = 4 s + z (0 , ℓ , n ) , then b ij = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s + z (0 , ℓ , n ) < j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (26) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z (0 ,ℓ ,n ) ,z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn . (7) If r = 6 , then Ω ( Y (26) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j + s ) → ( j + m +1) n + g ( j + s ) ⊗ e jn + g ( j + s ) . (8) If r = 7 , then Ω ( Y (26) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn +5 .
24 MARIYA PUSTOVYKH (9) If r = 8 , then Ω ( Y (26) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 . (10) If r = 9 , then Ω ( Y (26) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s )+1 . (11) If r = 10 , then Ω ( Y (26) t ) is described with (6 s × s ) -matrix with one nonzero elementthat is of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ e jn + g ( j + s ) . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (26) t ) is a Ω r ( Y (26) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 40 (Translates for the case 27) . (I) Let r ∈ N , r < . r -translates of theelements Y (27) t are described by the following way. (1) If r = 0 , then Ω ( Y (27) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 . (2) If r = 1 , then Ω ( Y (27) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j ) . (3) If r = 2 , then Ω ( Y (27) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) . (4) If r = 3 , then Ω ( Y (27) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z (0 ,ℓ ,n ) ,z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn . (5) If r = 4 , then Ω ( Y (27) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j )+1 . (6) If r = 5 , then Ω ( Y (27) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn +5 . (7) If r = 6 , then Ω ( Y (27) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j < s + z (0 , ℓ , n ) , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j = 3 s + z (0 , ℓ , n ) , then b ij = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s ) , i = j ;0 , otherwise.If s + z (0 , ℓ , n ) < j < s + z (0 , ℓ , n ) , then b ij = 0 .If j = 5 s + z (0 , ℓ , n ) , then b ij = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s )+1 , i = j ;0 , otherwise.If s + z (0 , ℓ , n ) < j < s , then b ij = 0 . (8) If r = 7 , then Ω ( Y (27) t ) is described with (8 s × s ) -matrix with one nonzero element thatis of the following form : b z (0 ,ℓ ,n ) ,z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn . (9) If r = 8 , then Ω ( Y (27) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) ,s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn + g ( j + s ) . (10) If r = 9 , then Ω ( Y (27) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j + s )+1 → ( j + m +1) n + g ( j + s )+1 ⊗ e jn +5 . (11) If r = 10 , then Ω ( Y (27) t ) is described with (6 s × s ) -matrix with one nonzero elementthat is of the following form : b s + z (0 ,ℓ ,n ) , s + z (0 ,ℓ ,n ) = − w ( j + m ) n + g ( j )+1 → ( j + m +1) n + g ( j )+1 ⊗ e jn + g ( j + s )+1 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (27) t ) is a Ω r ( Y (27) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ . Proposition 41 (Translates for the case 28) . (I) Let r ∈ N , r < . r -translates of theelements Y (28) t are described by the following way. (1) If r = 0 , then Ω ( Y (28) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b s, s = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn +5 . (2) If r = 1 , then Ω ( Y (28) t ) is described with (7 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If j = 4 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .
26 MARIYA PUSTOVYKH If j = 5 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (3) If r = 2 , then Ω ( Y (28) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b , = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn . (4) If r = 3 , then Ω ( Y (28) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If j = 2 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .If j = 3 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (5) If r = 4 , then Ω ( Y (28) t ) is described with (9 s × s ) -matrix with the following elements b ij : If j = 0 , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn , i = j ;0 , otherwise.If < j < s , then b ij = 0 .If j = 8 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn +5 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (6) If r = 5 , then Ω ( Y (28) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If j = 4 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .If j = 5 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j )+1 , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E (7) If r = 6 , then Ω ( Y (28) t ) is described with (9 s × s ) -matrix with one nonzero element thatis of the following form : b s, s = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn +5 . (8) If r = 7 , then Ω ( Y (28) t ) is described with (8 s × s ) -matrix with the following elements b ij : If j < s , then b ij = 0 .If j = 2 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 .If j = 3 s , then b ij = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If s < j < s , then b ij = 0 . (9) If r = 8 , then Ω ( Y (28) t ) is described with (6 s × s ) -matrix with one nonzero element thatis of the following form : b , = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn . (10) If r = 9 , then Ω ( Y (28) t ) is described with (7 s × s ) -matrix with one nonzero element thatis of the following form : b , = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn . (11) If r = 10 , then Ω ( Y (28) t ) is described with (6 s × s ) -matrix with one nonzero elementthat is of the following form : b s, s = w ( j + m ) n +5 → ( j + m +1) n +5 ⊗ e jn +5 . (II) Represent an arbitrary t ∈ N in the form t = 11 ℓ + r , where r . Then Ω t ( Y (28) t ) is a Ω r ( Y (28) t ) , whose left components twisted by σ ℓ , and coefficients multiplied by ( − ℓ .Proof. We prove the proposition 21, the other propositions are proved similarly. Show that thefollowing squares are commutative: Q t + t d t + t − −−−−→ Q t + t − f t y y f t − Q t d t − −−−→ Q t − . We shall describe the matrixes of products d t − f t and see that they coincide with f t − d t + t − .(1) If t <
11 and r = 0, then the product d r − f r is an (7 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < s + ( − ℓ ) s , then c ij = 0.
28 MARIYA PUSTOVYKH If j = 6 s + ( − ℓ ) s , then c ij = − κw ( j + m ) n → ( j + m +1) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + s ; − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ; − κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 3 s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ; − κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 5 s ;0 , otherwise.If 6 s + ( − ℓ ) s < j < s , then c ij = 0.(2) If t <
11 and r = 1, then the product d r − f r is an (6 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < s + ( − ℓ ) s , then c ij = 0.If j = s + ( − ℓ ) s , then c ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 4 s ;0 , otherwise.If s + ( − ℓ ) s < j < s + ( − ℓ ) s , then c ij = 0.If j = 2 s + ( − ℓ ) s , then c ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 3 s ;0 , otherwise.If 2 s + ( − ℓ ) s < j < s + ( − ℓ ) s , then c ij = 0.If j = 3 s + ( − ℓ ) s , then c ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If 3 s + ( − ℓ ) s < j < s + ( − ℓ ) s , then c ij = 0.If j = 4 s + ( − ℓ ) s , then c ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + s ;0 , otherwise.If 4 s + ( − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 5 s + ( − − ℓ ) s , then c ij = κw ( j + m +1) n +5 → ( j + m +2) n ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 5 s ;0 , otherwise.If 5 s + ( − − ℓ ) s < j < s , then c ij = 0.(3) If t <
11 and r = 2, then the product d r − f r is an (8 s × s )-matrix C = ( c ij ) with thefollowing elements c ij : OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If 0 j < ( − ℓ ) s , then c ij = 0.If j = ( − ℓ ) s , then c ij = w ( j + m − n + g ( j )+1 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + 2 s ; w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; − w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If ( − ℓ ) s < j < s + ( − ℓ ) s , then c ij = 0.If j = s + ( − ℓ ) s , then c ij = − w ( j + m − n + g ( j )+1 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j ) , i = j + 2 s ; w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 3 s ; − w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s + ( − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 6 s + ( − − ℓ ) s , then c ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 5 s ; w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 6 s ;0 , otherwise.If 6 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 7 s + ( − − ℓ ) s , then c ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ; w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn +5 → ( j +1) n +5 , i = ( j + 1) s + 6 s ;0 , otherwise.If 7 s + ( − − ℓ ) s < j < s , then c ij = 0.(4) If t <
11 and r = 3, then the product d r − f r is an (9 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < s + ( − − ℓ ) s , then c ij = 0.If j = 6 s + ( − − ℓ ) s , then c ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 3 s ;0 , otherwise.If 6 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.
30 MARIYA PUSTOVYKH If j = 7 s + ( − − ℓ ) s , then c ij = w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j )+1 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 4 s ;0 , otherwise.If 7 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 8 s + ( − − ℓ ) s , then c ij = κw ( j + m ) n +5 → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m +1) n + g ( j + s ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j )+1 , i = ( j + 1) s + 3 s ; κw ( j + m +1) n + g ( j ) → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s )+1 , i = ( j + 1) s + 4 s ;0 , otherwise.If 8 s + ( − − ℓ ) s < j < s , then c ij = 0.(5) If t <
11 and r = 4, then the product d r − f r is an (8 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < s + ( − − ℓ ) s , then c ij = 0.If j = 2 s + ( − − ℓ ) s , then c ij = κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If 2 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 3 s + ( − − ℓ ) s , then c ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ;0 , otherwise.If 3 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 4 s + ( − − ℓ ) s , then c ij = κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s + s ; κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 2 s ; − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If 4 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 5 s + ( − − ℓ ) s , then c ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn + g ( j )+1 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + s ; κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn + g ( j )+1 → jn +5 , i = j + 2 s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If 5 s + ( − − ℓ ) s < j < s , then c ij = 0.(6) If t <
11 and r = 5, then the product d r − f r is an (9 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < s + ( − − ℓ ) s , then c ij = 0.If j = s + ( − − ℓ ) s , then c ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 2 s + ( − − ℓ ) s , then c ij = − w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If 2 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 3 s + ( − − ℓ ) s , then c ij = w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ; − w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 5 s ;0 , otherwise.If 3 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 4 s + ( − − ℓ ) s , then c ij = − w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s ) → jn +5 , i = j + 4 s ;0 , otherwise.If 4 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 5 s + ( − − ℓ ) s , then c ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 3 s ;0 , otherwise.If 5 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 6 s + ( − − ℓ ) s , then c ij = − w ( j + m ) n +5 → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If 6 s + ( − − ℓ ) s < j < s , then c ij = 0.(7) If t <
11 and r = 6, then the product d r − f r is an (8 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < ( − − ℓ ) s , then c ij = 0.
32 MARIYA PUSTOVYKH If j = ( − − ℓ ) s , then c ij = w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → ( j +1) n , i = ( j + 1) s ; w ( j + m ) n → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ; − w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ; − w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 5 s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = s + ( − − ℓ ) s , then c ij = w ( j + m ) n + g ( j ) → ( j + m ) n + g ( j )+1 ⊗ w jn → ( j +1) n , i = ( j + 1) s + s ; w ( j + m ) n → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s ) , i = j + s ; − w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j + s )+1 , i = j + 3 s ; − w ( j + m − n +5 → ( j + m ) n + g ( j )+1 ⊗ w jn → jn + g ( j )+1 , i = j + 4 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 2 s + ( − − ℓ ) s , then c ij = − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If 2 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 3 s + ( − − ℓ ) s , then c ij = − κw ( j + m ) n + g ( j ) → ( j + m ) n +5 ⊗ w jn + g ( j ) → ( j +1) n , i = ( j + 1) s + s ; κw ( j + m ) n → ( j + m ) n +5 ⊗ e jn + g ( j ) , i = j ;0 , otherwise.If 3 s + ( − − ℓ ) s < j < s , then c ij = 0.(8) If t <
11 and r = 7, then the product d r − f r is an (6 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < s + ( − − ℓ ) s , then c ij = 0.If j = s + ( − − ℓ ) s , then c ij = − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If j = 2 s + ( − − ℓ ) s , then c ij = − w ( j + m +1) n → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; w ( j + m +1) n + g ( j + s ) → ( j + m +1) n + g ( j + s )+1 ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise.If 2 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 5 s + ( − − ℓ ) s , then c ij = κw ( j + m +1) n → ( j + m +2) n ⊗ w jn +5 → ( j +1) n , i = ( j + 1) s ; − κw ( j + m +1) n + g ( j + s ) → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; − κw ( j + m +1) n + g ( j ) → ( j + m +2) n ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ;0 , otherwise.If 5 s + ( − − ℓ ) s < j < s , then c ij = 0.(9) If t <
11 and r = 8, then the product d r − f r is an (7 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < ( − − ℓ ) s , then c ij = 0.If j = ( − − ℓ ) s , then c ij = κw ( j + m ) n + g ( j )+1 → ( j + m ) n +5 ⊗ w jn → ( j +1) n , i = ( j + 1) s ; − κw ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ⊗ w jn → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = s + ( − − ℓ ) s , then c ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s ; − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn +5 , i = j + 5 s ; − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn +5 , i = j + 6 s ;0 , otherwise.If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 2 s + ( − − ℓ ) s , then c ij = − κw ( j + m ) n + g ( j + s )+1 → ( j + m +1) n ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ; − κw ( j + m ) n + g ( j ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn +5 , i = j + 4 s ; − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn + g ( j + s ) → jn +5 , i = j + 5 s ;0 , otherwise.If 2 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.
34 MARIYA PUSTOVYKH If j = 3 s + ( − − ℓ ) s , then c ij = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 4 s ;0 , otherwise.If 3 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 4 s + ( − − ℓ ) s , then c ij = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise.If 4 s + ( − − ℓ ) s < j < s , then c ij = 0.(10) If t <
11 and r = 9, then the product d r − f r is an (6 s × s )-matrix C = ( c ij ) with thefollowing elements c ij :If 0 j < ( − − ℓ ) s , then c ij = 0.If j = ( − − ℓ ) s , then c ij = κw ( j + m ) n +5 → ( j + m +1) n ⊗ w jn → ( j +1) n , i = ( j + 1) s ; κw ( j + m ) n + g ( j )+1 → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + s ; − κw ( j + m ) n + g ( j + s ) → ( j + m +1) n ⊗ w jn → jn + g ( j )+1 , i = j + 3 s ; κe ( j + m +1) n ⊗ w jn → jn +5 , i = j + 5 s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 2 s + ( − − ℓ ) s , then c ij = w ( j + m ) n + g ( j ) → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → jn + g ( j + s )+1 , i = j + 2 s ; − w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → jn +5 , i = j + 3 s ;0 , otherwise.If 2 s + ( − − ℓ ) s < j < s , then c ij = 0.(11) If t <
11 and r = 10, then the product d r − f r is an (6 s × s )-matrix C = ( c ij ) withthe following elements c ij :If 0 j < ( − − ℓ ) s , then c ij = 0.If j = ( − − ℓ ) s , then c ij = κw ( j + m ) n → ( j + m +1) n ⊗ w jn → jn + g ( j ) , i = j + s ; − κw ( j + m ) n → ( j + m +1) n ⊗ w jn → jn + g ( j + s ) , i = j + 2 s ;0 , otherwise.If ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = s + ( − − ℓ ) s , then c ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise. OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E If s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 2 s + ( − − ℓ ) s , then c ij = w ( j + m +1) n → ( j + m +1) n + g ( j ) ⊗ w jn + g ( j + s ) → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise.If 2 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 3 s + ( − − ℓ ) s , then c ij = − w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ;0 , otherwise.If 3 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 4 s + ( − − ℓ ) s , then c ij = − w ( j + m +1) n → ( j + m +1) n + g ( j )+1 ⊗ w jn + g ( j + s )+1 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + 2 s ;0 , otherwise.If 4 s + ( − − ℓ ) s < j < s + ( − − ℓ ) s , then c ij = 0.If j = 5 s + ( − − ℓ ) s , then c ij = − κw ( j + m +1) n → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j + s ) , i = ( j + 1) s + s ; κw ( j + m +1) n → ( j + m +1) n +5 ⊗ w jn +5 → ( j +1) n + g ( j ) , i = ( j + 1) s + 2 s ;0 , otherwise.If 5 s + ( − − ℓ ) s < j < s , then c ij = 0.(12) If t >
11, then the product matrix d t − f t is a d r − f r , whose left components twistedby σ ℓ , and coefficients multiplied by ( − ℓ .The described matrixes of products d t − f t coincide with f t − d t + t − . (cid:3) Multiplications in HH ∗ ( R )From the descriptions of elements Y ( i ) t and itsΩ-translates we can find multiplications of theelements using the formula ( ∗ ).To find the multiplications we need the description of σ t for an arbitrary t ∈ N . From thedescription of an automorphism σ we have: σ t ( α i ) = ( − t α i +6 tn , i ≡ , α i +6 tn , i ≡ , σ t ( β i ) = ( − t β i +6 tn , i ≡ , β i +6 tn , i ≡ ,σ t +1 ( α i ) = ( − t +1 β i +3(2 t +1) n , i ≡ − β i +3(2 t +1) n , i ≡ − t β i +3(2 t +1) n , i ≡ , σ t +1 ( β i ) = ( − t α i +3(2 t +1) n , i ≡ − α i +3(2 t +1) n , i ≡ − t +1 α i +3(2 t +1) n , i ≡ ,σ t ( γ i ) = ( − t γ i + tn . We will find a multiplication of elements of the types 4 and 3 for s >
36 MARIYA PUSTOVYKH
Consider two arbitrary elements Y (4) t and Y (3) t . For its degrees t and t we have: t = 11 ℓ + 1 , ℓ n ≡ s ) , ℓ ... 2; t = 11 ℓ + 1 , ℓ n ≡ s ) , ℓ ... 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).Denote by B = ( b ij ) translate matrix of an element Y (4) t by degree t . This matrix is of theform.If 0 j < s , then b ij = κσ ℓ ( w ( j + m ) n + g ( j + s ) → ( j + m ) n +5 ) ⊗ e jn , i = j + s ; κσ ℓ ( w ( j + m ) n + g ( j + s )+1 → ( j + m ) n +5 ) ⊗ w jn → jn + g ( j + s ) , i = j + 3 s ;0 , otherwise;= κσ ℓ ( w ( j + m ) n +3 → ( j + m ) n +5 ) ⊗ e jn , i = j + s ; κσ ℓ ( w ( j + m ) n +4 → ( j + m ) n +5 ) ⊗ w jn → jn +3 , i = j + 3 s ;0 , otherwise;= κσ ℓ ( β j + m )+2 β j + m )+1 ) ⊗ e jn , i = j + s ; κσ ℓ ( β j + m )+2 ) ⊗ β j , i = j + 3 s ;0 , otherwise;= ( − ℓ − ( − ℓ β j + ℓ n )+2 β j + ℓ n )+1 ⊗ e jn , i = j + s ;( − ℓ − ( − ℓ β j + ℓ n )+2 ⊗ β j , i = j + 3 s ;0 , otherwise;= ( − ℓ ℓ − β j +2 β j +1 ⊗ e jn , i = j + s ;( − ℓ ℓ − β j +2 ⊗ β j , i = j + 3 s ;0 , otherwise.If s j < s , then b ij = σ ℓ ( w ( j + m +1) n + g ( j ) → ( j + m +1) n + g ( j )+1 ) ⊗ w jn + g ( j + s ) → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise;= σ ℓ ( w ( j + m +1) n +3 → ( j + m +1) n +4 ) ⊗ w jn +1 → ( j +1) n , i = ( j + 1) s + s ;0 , otherwise; OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E = σ ℓ ( β j + m +1)+1 ) ⊗ γ j α j +2 α j +1 , i = ( j + 1) s + s ;0 , otherwise;= β j +1)+1 ⊗ γ j α j +2 α j +1 , i = ( j + 1) s + s ;0 , otherwise.If 2 s j < s , then b ij = 0.If 4 s j < s , then b ij = σ ℓ ( w ( j + m ) n +5 → ( j + m +1) n + g ( j + s ) ) ⊗ e jn + g ( j + s )+1 , i = j + s ; σ ℓ ( w ( j + m +1) n → ( j + m +1) n + g ( j + s ) ) ⊗ w jn + g ( j + s )+1 → jn +5 , i = j + 2 s ;0 , otherwise;= σ ℓ ( β j + m +1) γ j + m ) ⊗ e jn +4 , i = j + s ; σ ℓ ( β j + m +1) ) ⊗ β j +2 , i = j + 2 s ;0 , otherwise;= ( − ℓ β j +1) γ j ⊗ e jn +4 , i = j + s ;( − ℓ β j +1) ⊗ β j +2 , i = j + 2 s ;0 , otherwise.If 5 s j < s , then b ij = κσ ℓ ( w ( j + m +1) n → ( j + m +2) n ) ⊗ e jn +5 , i = j + s ;0 , otherwise;= κσ ℓ ( γ j + m +1 α j + m +1)+2 α j + m +1)+1 α j + m +1) ) ⊗ e jn +5 , i = j + s ;0 , otherwise;= ( − ℓ − γ j +1 α j +1)+2 α j +1)+1 α j +1) ⊗ e jn +5 , i = j + s ;0 , otherwise.Multiply this matrix by an element Y (3) t , which is (7 s × s )-matrix with single nonzero element y s, s = ( − ℓ γ j ⊗ e jn +5 . Multiplication is the matrix C = ( c ij ) with single nonzero element c s, s = β j +1) γ j ⊗ β j +2 . We must show that this element coincide with Y (5) t for degree of type (5). Element Y (5) t ismatrix with single nonzero element y s, s = α j +1) γ j α j +2 ⊗ e jn +2 . Thus we must show that Y (5) t − C = α j +1) γ j α j +2 − β j +1) γ j β j +2 is in Im δ t − .Consider the matrix D = ( d ) ij , which is obtained from differential matrix d t − by replacingelements which turns into 0 in Im δ t − , by 0. Elements of this matrix are of the form.
38 MARIYA PUSTOVYKH
If 0 j < s , then( d ) ij = σ ℓ + ℓ ( w jn +1+ j +2 f ( j , → jn +5 ) ⊗ w jn → jn + j ,i = j + 2 sj , j < − σ ℓ + ℓ ( w jn +3+ j → jn +5 ) ⊗ w jn → jn + j +2(1 − f ( j , ,i = j + 2 sj + s, j < ( − ℓ ℓ β j +2 β j +1 ⊗ e jn , i = j ; e jn +5 ⊗ α j +1 α j , i = j + 4 s ;( − ℓ ℓ α j +2 α j +1 ⊗ e jn , i = j + s ; − e jn +5 ⊗ β j +1 β j , i = j + 5 s ;0 otherwise.If s j < s , then ( d ) ij = 0.If 3 s j < s , then( d ) ij = ( − ℓ ℓ α j +1) γ j ⊗ e jn +1 , i = j + s, j < s ;( − ℓ ℓ − β j +1) γ j ⊗ e jn +4 , i = j + s, j > s ;( − ℓ ℓ − α j +1) ⊗ α j +1 , i = ( j ) s + 6 s, j < s ;( − ℓ ℓ β j +1) ⊗ β j +1 , i = ( j ) s + 6 s, j > s ; e ( j +1) n + g ( j ) ⊗ w (2) jn + g ( j + s )+1 → , i = ( j + 1) s + sf ( j, s );0 otherwise.If 5 s j < s , then ( d ) ij = 0.6 s -th row of image matrix multiplied by ( − ℓ ℓ − , coincide with Y (5) t − C , hence, Y (5) t − Y (4) t Y (3) t ∈ Im δ t − , i. e. Y (5) t − Y (4) t Y (3) t = 0 in the cohomology ring.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 42. ( 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) . ( 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) OCHSCHILD COHOMOLOGY RING FOR SELF-INJECTIVE ALGEBRAS OF TREE CLASS E References [1] C. Riedtmann,
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