One-generated nilpotent assosymmetric algebras
aa r X i v : . [ m a t h . R A ] J a n One-generated nilpotent assosymmetric algebras
Ivan Kaygorodov & Farukh Mashurov
E-mail addresses:Ivan Kaygorodov ([email protected])Farukh Mashurov ([email protected])
Abstract : We give the classification of - and -dimensional complex one-generated nilpotent as-sosymmetric algebras. Keywords : assosymmetric algebras, nilpotent algebras, algebraic classification, central extension. MSC2010 : 17A30, 17D25. I
NTRODUCTION
Algebraic classification (up to isomorphism) of algebras of small dimension from a certain vari-ety defined by a family of polynomial identities is a classic problem in the theory of non-associativealgebras. There are many results related to algebraic classification of small dimensional algebras invarieties of Jordan, Lie, Leibniz, Zinbiel and other algebras. Another interesting approach of study-ing algebras of a fixed dimension is to study them from a geometric point of view (that is, to studydegenerations and deformations of these algebras). The results in which the complete informationabout degenerations of a certain variety is obtained are generally referred to as the geometric clas-sification of the algebras of these variety. There are many results related to geometric classificationof Jordan, Lie, Leibniz, Zinbiel and other algebras [1, 5, 24, 35]. Another interesting direction is astudy of one-generated objects. Well know the description of one-generated finite groups: there isonly one one-generated group of order n . In the case of algebras, there are some similar results,such that the description of n -dimensional one-generated nilpotent associative [14], noncommutativeJordan [25], Leibniz and Zinbiel algebras [38]. It was proven, that there is only one n -dimensionalone-generated nilpotent algebra in these varieties. But on the other side, as we can see in varietiesof Novikov [27], assosymmetric [23], bicommutative [33], commutative [17], and terminal [30] al-gebras, there are more than one -dimensional one-generated nilpotent algebra from these varieties.One-generated nilpotent Novikov algebras in dimensions 5 and 6 were studied in [8], one-generatednilpotent terminal algebras in dimension 5 were studied in [31]. In the present paper, we give the alge-braic classification of - and -dimensional complex one-generated nilpotent assosymmetric algebras,which first appeared in the paper by Kleinfeld in 1957 [37].The variety of assosymmetric algebras is defined by the following identities of right- and left-symmetric: ( x, y, z ) = ( x, z, y ) , ( x, y, z ) = ( y, x, z ) , where ( x, y, z ) = ( xy ) z − x ( yz ) . It admits the commutative associative and associative algebras asa subvariety. Kleinfeld proved that an assosymmetric ring of characteristic different from 2 and 3without ideals I = 0 , such that I = 0 is associative [37]. The free base elements of assosymmetricalgebras were described in [22]. The algebraic and geometric classification of -dimensional complexnilpotent assosymmetric algebras was given in [23]. Also, assosymmetric algebras were studied in[2, 3, 15, 16, 36, 39].The key step in our method for algebraically classifying assosymmetric nilpotent algebras is thecalculation of central extensions of smaller algebras. It comes as no surprise that the central exten-sions of Lie and non-Lie algebras have been exhaustively studied for years. It is interesting both todescribe them and to use them to classify different varieties of algebras [7, 9, 20, 28, 32, 32, 34, 40].Firstly, Skjelbred and Sund devised a method for classifying nilpotent Lie algebras employing centralextensions [40]. Using this method, all the non-Lie central extensions of all -dimensional Malcevalgebras were described afterwards [20], and also all the anticommutative central extensions of the -dimensional anticommutative algebras [4], and all the central extensions of the -dimensional alge-bras [6]. Moreover, the method is especially indicated for the classification of nilpotent algebras andit was used to describe all the -dimensional nilpotent associative algebras [13], all the -dimensionalnilpotent Novikov algebras [27], all the -dimensional nilpotent bicommutative algebras [33], allthe -dimensional nilpotent Jordan algebras [19], all the -dimensional nilpotent restricted Lie alge-bras [11], all the -dimensional nilpotent Lie algebras [10,12], all the -dimensional nilpotent Malcevalgebras [21], all the -dimensional nilpotent Tortkara algebras [18] and some others.1. T HE ALGEBRAIC CLASSIFICATION OF NILPOTENT ASSOSYMMETRIC ALGEBRAS
Method of classification of nilpotent algebras.
The objective of this section is to give an ana-logue of the Skjelbred-Sund method for classifying nilpotent assosymmetric algebras. As other ana-logues of this method were carefully explained in, for example, [6, 20], we will give only someimportant definitions, and refer the interested reader to the previous sources. We will also employtheir notations.Let ( A , · ) be an assosymmetric algebra over C and V a vector space over C . We define the C -linearspace Z ( A , V ) as the set of all bilinear maps θ : A × A −→ V such that θ ( xy, z ) − θ ( x, yz ) = θ ( xz, y ) − θ ( x, zy ) ,θ ( xy, z ) − θ ( x, yz ) = θ ( yx, z ) − θ ( y, xz ) . These maps will be called cocycles . Consider a linear map f from A to V , and set δf : A × A −→ V with δf ( x, y ) = f ( xy ) . Then, δf is a cocycle, and we define B ( A , V ) = { θ = δf : f ∈ Hom ( A , V ) } , a linear subspace of Z ( A , V ) ; its elements are called coboundaries .The second cohomology space H ( A , V ) is defined to be the quotient space Z ( A , V ) (cid:14) B ( A , V ) .Let Aut( A ) be the automorphism group of the assosymmetric algebra A and let φ ∈ Aut( A ) .Every θ ∈ Z ( A , V ) defines φθ ( x, y ) = θ ( φ ( x ) , φ ( y )) , with φθ ∈ Z ( A , V ) . It is easily checked that Aut( A ) acts on Z ( A , V ) , and that B ( A , V ) is invariant under the action of Aut( A ) . So, wehave that
Aut( A ) acts on H ( A , V ) .Let A be an assosymmetric algebra of dimension m < n over C , V a C -vector space of dimension n − m and θ a cocycle, and consider the direct sum A θ = A ⊕ V with the bilinear product “ [ − , − ] A θ ”defined by [ x + x ′ , y + y ′ ] A θ = xy + θ ( x, y ) for all x, y ∈ A , x ′ , y ′ ∈ V . It is straightforward that A θ is an assosymmetric algebra if and only if θ ∈ Z ( A , V ) ; it is called an ( n − m ) - dimensional centralextension of A by V .We also call the set Ann( θ ) = { x ∈ A : θ ( x, A ) + θ ( A , x ) = 0 } the annihilator of θ . We recallthat the annihilator of an algebra A is defined as the ideal Ann( A ) = { x ∈ A : x A + A x = 0 } .Observe that Ann ( A θ ) = (cid:0) Ann( θ ) ∩ Ann( A ) (cid:1) ⊕ V . Definition 1.
Let A be an algebra and I be a subspace of Ann( A ) . If A = A ⊕ I then I is calledan annihilator component of A . A central extension of an algebra A without annihilator componentis called a non-split central extension. The following result is fundamental for the classification method.
Lemma 2.
Let A be an n -dimensional assosymmetric algebra such that dim Ann( A ) = m = 0 .Then there exists, up to isomorphism, a unique ( n − m ) -dimensional assosymmetric algebra A ′ anda bilinear map θ ∈ Z ( A , V ) with Ann( A ) ∩ Ann( θ ) = 0 , where V is a vector space of dimensionm, such that A ∼ = A ′ θ and A / Ann( A ) ∼ = A ′ . For the proof, we refer the reader to [20, Lemma 5].Now, we seek a condition on the cocycles to know when two ( n − m ) -central extensions areisomorphic. Let us fix a basis e , . . . , e s of V , and θ ∈ Z ( A , V ) . Then θ can be uniquely writtenas θ ( x, y ) = s X i =1 θ i ( x, y ) e i , where θ i ∈ Z ( A , C ) . It holds that θ ∈ B ( A , V ) if and only if all θ i ∈ B ( A , C ) , and it also holds that Ann( θ ) = Ann( θ ) ∩ Ann( θ ) . . . ∩ Ann( θ s ) . Furthermore, if Ann( θ ) ∩ Ann ( A ) = 0 , then A θ has an annihilator component if and only if [ θ ] , [ θ ] , . . . , [ θ s ] arelinearly dependent in H ( A , C ) (see [20, Lemma 13]).Recall that, given a finite-dimensional vector space V over C , the Grassmannian G k ( V ) is the setof all k -dimensional linear subspaces of V . Let G s (H ( A , C )) be the Grassmannian of subspacesof dimension s in H ( A , C ) . For W = h [ θ ] , [ θ ] , . . . , [ θ s ] i ∈ G s (H ( A , C )) and φ ∈ Aut( A ) ,define φ W = h [ φθ ] , [ φθ ] , . . . , [ φθ s ] i . It holds that φ W ∈ G s (H ( A , C )) , and this induces anaction of Aut( A ) on G s (H ( A , C )) . We denote the orbit of W ∈ G s (H ( A , C )) under this actionby Orb(W) . Let W = h [ θ ] , [ θ ] , . . . , [ θ s ] i , W = h [ ϑ ] , [ ϑ ] , . . . , [ ϑ s ] i ∈ G s (cid:0) H ( A , C ) (cid:1) . Similarly to [20, Lemma 15], in case W = W , it holds that s \ i =1 Ann( θ i ) ∩ Ann ( A ) = s \ i =1 Ann( ϑ i ) ∩ Ann( A ) , and therefore the set T s ( A ) = ( W = h [ θ ] , [ θ ] , . . . , [ θ s ] i ∈ G s (cid:0) H ( A , C ) (cid:1) : s \ i =1 Ann( θ i ) ∩ Ann( A ) = 0 ) is well defined, and it is also stable under the action of Aut( A ) (see [20, Lemma 16]).Now, let V be an s -dimensional linear space and let us denote by E ( A , V ) the set of all non-split s -dimensional central extensions of A by V . We can write E ( A , V ) = ( A θ : θ ( x, y ) = s X i =1 θ i ( x, y ) e i and h [ θ ] , [ θ ] , . . . , [ θ s ] i ∈ T s ( A ) ) . Finally, we are prepared to state our main result, which can be proved as [20, Lemma 17].
Lemma 3.
Let A θ , A ϑ ∈ E ( A , V ) . Suppose that θ ( x, y ) = s X i =1 θ i ( x, y ) e i and ϑ ( x, y ) = s X i =1 ϑ i ( x, y ) e i . Then the assosymmetric algebras A θ and A ϑ are isomorphic if and only if Orb h [ θ ] , [ θ ] , . . . , [ θ s ] i = Orb h [ ϑ ] , [ ϑ ] , . . . , [ ϑ s ] i . Then, it exists a bijective correspondence between the set of
Aut( A ) -orbits on T s ( A ) and the setof isomorphism classes of E ( A , V ) . Consequently we have a procedure that allows us, given anassosymmetric algebra A ′ of dimension n − s , to construct all non-split central extensions of A ′ .ProcedureLet A ′ be an assosymmetric algebra of dimension n − s .(1) Determine H ( A ′ , C ) , Ann( A ′ ) and Aut( A ′ ) .(2) Determine the set of Aut( A ′ ) -orbits on T s ( A ′ ) .(3) For each orbit, construct the assosymmetric algebra associated with a representative of it.1.2. Notations.
Let A be an assosymmetric algebra with a basis e , e , . . . , e n . Then by ∆ ij we willdenote the assosymmetric bilinear form ∆ ij : A × A −→ C with ∆ ij ( e l , e m ) = δ il δ jm . Then the set { ∆ ij : 1 ≤ i, j ≤ n } is a basis for the linear space of the bilinear forms on A . Then every θ ∈ Z ( A ) can be uniquely written as θ = X ≤ i,j ≤ n c ij ∆ ij , where c ij ∈ C . Let us fix the following notations: A ij — j th i -dimensional one-generated nilpotent assosymmetric algebra. The algebraic classification of low dimensional one-generated nilpotent assosymmetric al-gebras.
In the present table (thanks to [23]) we have a description of all -, - and -dimensionalone-generated nilpotent assosymmetric algebras: A : e e = e A : e e = e e e = e A ( α ) : e e = e e e = e e e = αe A : e e = e e e = e e e = e A : e e = e e e = e e e = e e e = e e e = − e e e = − e A : e e = e e e = e e e = e e e = − e e e = − e A ( α ) : e e = e e e = e e e = (2 − α ) e e e = αe e e = ( α − α + 1) e e e = (2 α − e A : e e = e e e = e e e = − e e e = e e e = − e e e = e A : e e = e e e = e e e = − e e e = − e + e e e = − e e e = 3 e Remark 4.
Note that, non-split central extension of a split algebra can not be a one-generated alge-bra. Hence, we will consider central extensions only for non-split one-generated nilpotent algebras.
2. C
LASSIFICATION OF - DIMENSIONAL ONE - GENERATED NILPOTENT ASSOSYMMETRICALGEBRAS -dimensional central extensions of -dimensional one-generated algebras. The second co-homology spaces of algebras A , A ( α ) given in [23]. Therefore, two dimensional central exten-sions of these algebras gives the following: A : e e = e e e = e e e = e e e = e e e = − e e e = − e A ( α ) : e e = e e e = e e e = ( α − e e e = αe + e e e = ( α − α − e e e = (1 − α ) e Cohomology spaces of -dimensional one-generated assosymmetric algebras. In the presenttable we collect all usefull information about Z , B and H spaces for all -dimensional one-generated algebras that were counted via code in [26]. Z ( A ) = h ∆ , ∆ , ∆ , ∆ + ∆ , ∆ − ∆ − ∆ , ∆ + 2∆ + ∆ i B ( A ) = h ∆ , ∆ , ∆ i H ( A ) = h [∆ ] + [∆ ] , [∆ ] − [∆ ] − [∆ ] , [∆ ] + 2[∆ ] + [∆ ] i Z ( A ) = h ∆ , ∆ , ∆ , ∆ − ∆ − i B ( A ) = h ∆ , ∆ , ∆ + ∆ − ∆ − i H ( A ) = h [∆ ] i Z ( A ) = h ∆ , ∆ , ∆ , ∆ − ∆ − i B ( A ) = h ∆ , ∆ , ∆ − ∆ − i H ( A ) = h [∆ ] i Z ( A ( α ) α =1 ) = h ∆ , ∆ , ∆ , (2 − α )∆ + ( α − α + 1)∆ + (2 α − i B ( A ( α ) α =1 ) = h ∆ , ∆ + α ∆ , (2 − α )∆ + ( α − α + 1)∆ + (2 α − i H ( A ( α ) α =1 ) = h [∆ ] i Z ( A (1)) = h ∆ , ∆ , ∆ , ∆ + ∆ + ∆ , ∆ + ∆ + ∆ + ∆ i B ( A (1)) = h ∆ , ∆ + ∆ , ∆ + ∆ + ∆ i H ( A (1)) = h [∆ ] , [∆ ] + [∆ ] + [∆ ] + [∆ ] i Z ( A ) = h ∆ , ∆ , ∆ , + ∆ − ∆ i B ( A ) = h ∆ , ∆ , − + ∆ − ∆ + ∆ i H ( A ) = h [∆ ] i Z ( A ) = h ∆ , ∆ , ∆ , ∆ + ∆ − ∆ i B ( A ) = h ∆ , ∆ − ∆ , − + ∆ − + 3∆ i H ( A ) = h [∆ ] + [∆ ] − [∆ ] i Remark 5.
Extensions of the algebras A , A , A ( α ) α =1 , A and A give algebras with -dimensional annihilator. Then, in the following subsections we study the central extensions of theother algebras. Central extensions of A . Let us use the following notations: ∇ = [∆ ] + [∆ ] , ∇ = [∆ ] − [∆ ] − [∆ ] , ∇ = [∆ ] + 2[∆ ] + [∆ ] . The automorphism group of A consists of invertible matrices of the form φ = x y x z xy x t xy x . Since φ T α α α − α + 2 α α − α + α φ = α ∗ α ∗∗ α ∗ α ∗ α ∗∗∗ α ∗ − α ∗ + 2 α ∗ α ∗ − α ∗ + α ∗ , we have that the action of Aut( A ) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α , α ∗ = x α , α ∗ = x α . -dimensional central extensions. We have the following new cases:(1) If α = 0 , α = 0 , α = 0 , then x = √ α , we have the representative h∇ i ; (2) If α = 0 , α = 0 , then x = √ α , α = α α we have the representative h α ∇ + ∇ i ; (3) If α = 0 , then x = √ α , α = α α , β = α α we have the representative h α ∇ + β ∇ + ∇ i . From here, we have new -dimensional one generated assosymmetric algebras constructed from A : A : e e = e e e = e e e = e e e = e e e = e A ( α ) : e e = e e e = e e e = e e e = αe e e = e e e = − e e e = ( α − e A ( α, β ) : e e = e e e = e e e = e e e = αe e e = βe e e = e e e = (2 − β ) e e e = ( α − β + 1) e -dimensional central extensions. Consider the vector space generated by the following twococycles θ = α ∇ + α ∇ + α ∇ ,θ = β ∇ + β ∇ . Here we have the following cases:(1) If α = 0 , then we have the representative h∇ , ∇ i ; (2) If α = 0 , β = 0 , β = 0 , then we have the representative h∇ , α ∇ + ∇ i ; (3) If α = 0 , β = 0 , then we have the representative h α ∇ + ∇ , β ∇ + ∇ i . We have the following new -dimensional one-generated nilpotent assosymmetric algebras con-structed from A : A : e e = e e e = e e e = e e e = e e e = e e e = − e e e = e − e A ( α ) : e e = e e e = e e e = e e e = αe e e = e e e = e e e = (2 − α ) e e e = e − ( α − e A ( α, β ) : e e = e e e = e e e = αe + βe e e = e e e = e e e = e e e = − e + 2 e e e = ( α − e + ( β + 1) e Central extensions of A (1) . Let us use the following notations: ∇ = [∆ ] , ∇ = [∆ ] + [∆ ] + [∆ ] + [∆ ] . The automorphism group of A (1) consists of invertible matrices of the form φ = x y x z xy x t xz + y yx x . Since φ T α α α α α φ = α ∗∗∗ α ∗ α ∗∗ α ∗ α ∗ + α ∗ α ∗∗ α ∗ α ∗∗ α ∗ α ∗ , we have that the action of Aut( A (1)) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α , α ∗ = x α . -dimensional central extensions. Note that if α = 0 then we obtain algebras with 2-dimensional annihilator. Therefore, we have two representatives h∇ i and h∇ + ∇ i dependingon whether α = 0 or not.We have the following new -dimensional nilpotent assosymmetric algebras constructed from A (1) : A : e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e A : e e = e e e = e e e = e e e = e + e e e = e e e = e e e = e e e = e e e = e e e = e -dimensional central extensions. We have only one new -dimensional nilpotent assosym-metric algebras constructed from A (1) : A : e e = e e e = e e e = e e e = e e e = e + e e e = e e e = e e e = e e e = e e e = e Classification theorem.
Summarizing results of the previous sections, we have the followingtheorem.
Theorem A.
Let A be a -dimensional complex one-generated nilpotent assosymmetric algebra, then A is isomorphic to an algebra from the following list: A e e = e e e = e e e = e e e = e e e = − e e e = − e A ( α ) e e = e e e = e e e = ( α − e e e = αe + e e e = ( α − α − e e e = (1 − α ) e A e e = e e e = e e e = e e e = e e e = e A ( α ) e e = e e e = e e e = αe e e = e e e = e e e = − e e e = ( α − e A ( α, β ) e e = e e e = e e e = αe e e = βe e e = e e e = e e e = (2 − β ) e e e = ( α − β + 1) e A e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e A e e = e e e = e e e = e e e = e e e = e + e e e = e e e = e e e = e e e = e e e = e
3. C
LASSIFICATION OF DIMENSIONAL ONE - GENERATED NILPOTENT ASSOSYMMETRICALGEBRAS
Cohomology spaces of -dimensional one-generated assosymmetric algebras. All multipli-cation tables of -dimensional one-generated nilpotent assosymmetric algebras is given in table inTheorem A (see, previous section). All necessary information about coboundaries, cocycles andsecond cohomology spaces of -dimensional one-generated nilpotent assosymmetric algebras werecalculated by the code in [26] and given in the following table. Table B. The list of cohomology spaces of 5-dimensional one-generated assosymmetric algebras Z ( A ) = D ∆ , ∆ , ∆ , ∆ + ∆ , ∆ + 2∆ + ∆ ∆ − ∆ − ∆ E B ( A ) = D ∆ , ∆ , ∆ , ∆ − ∆ − E H ( A ) = D [∆ ] + [∆ ] , [∆ ] − [∆ ] + [∆ ] E Z ( A ( α = 1)) = D ∆ , ∆ , ∆ , ∆ + (1 − α )∆ + (1 − α )∆ , ∆ − ∆ − , ∆ + ∆ + (2 − α )∆ E B ( A ( α = 1)) = D ∆ , ∆ , ∆ , ( α − + ( α − α − + (1 − α )∆ E H ( A ( α = 1)) = D [∆ ] − [∆ ] − ] , [∆ ] + [∆ ] + (2 − α )[∆ ] E Z ( A (1)) = D ∆ , ∆ , ∆ , ∆ − ∆ , ∆ + ∆ + ∆ , ∆ + ∆ − ∆ , ∆ − ∆ − ∆ + ∆ E B ( A (1)) = D ∆ , ∆ , ∆ , ∆ + ∆ + ∆ E H ( A (1)) = D [∆ ] − [∆ ] , [∆ ] + [∆ ] − [∆ ] , [∆ ] − [∆ ] − [∆ ] + [∆ ] E Z ( A ) = D ∆ , ∆ , ∆ , ∆ + ∆ , ∆ − ∆ − ∆ , ∆ + 2∆ + ∆ E B ( A ) = D ∆ , ∆ , ∆ , ∆ + ∆ E H ( A ) = D [∆ ] − [∆ ] − [∆ ] , [∆ ] + 2[∆ ] + [∆ ] E Z ( A ( α )) = D ∆ , ∆ , ∆ , ∆ + ∆ − ∆ , ∆ − ∆ − ∆ , ∆ + ∆ + ∆ E B ( A ( α )) = D ∆ , ∆ , ∆ , α ∆ + ∆ − ∆ + ( α − E H ( A ( α )) = D [∆ ] + [∆ ] , α [∆ ] + 2[∆ ] + [∆ ] + ( α − ] E Z ( A ( α, β )) = D ∆ , ∆ , ∆ , ∆ + ∆ − ∆ , ∆ − ∆ − ∆ , ∆ + ∆ + ∆ E B ( A ( α, β )) = D ∆ , ∆ , ∆ , α ∆ + β ∆ + ∆ + (2 − β )∆ + ( α − β + 1)∆ E H ( A ( α, β )) = D [∆ ] + [∆ ] − [∆ ] , [∆ ] − [∆ ] − [∆ ] E α = ( β ± p − β − β )Z ( A ( α, β )) = D ∆ , ∆ , ∆ , ∆ − ∆ − ∆ , ∆ + ∆ , (2 β − ++(2 β ( α −
1) + 1)∆ + ( α + 2 β − β + 1)∆ + ( − αβ + 3 α + 2 β − β + 1)∆ ++( − αβ + 2 α + 2 β − + (2 α − β + 1)∆ , ∆ + 2∆ + ∆ E B ( A ( α, β )) = D ∆ , ∆ , ∆ , α ∆ + β ∆ + ∆ + (2 − β )∆ + ( α − β + 1)∆ E H ( A ( α, β )) = D [∆ ] − [∆ ] − [∆ ] , [∆ ] + [∆ ] , (2 β − ] + (2 β ( α −
1) + 1)[∆ ] + ( α + 2 β − β + 1)[∆ ]+( − αβ + 3 α + 2 β − β + 1)[∆ ] + ( − αβ + 2 α + 2 β − ] + (2 α − β + 1)[∆ ] E α = ( β ± p − β − β ) and ( α, β ) = (0 , )Z ( A (0 , )) = D ∆ , ∆ , ∆ , ∆ + ∆ , ∆ − ∆ − ∆ , − − − + ∆ − , ∆ + 2∆ + ∆ E B ( A (0 , )) = D ∆ , ∆ , ∆ , ∆ + 2∆ + 3∆ + ∆ E H ( A (0 , )) = D [∆ ] − [∆ ] − [∆ ] , [∆ ] + [∆ ] , ] − ] − ] − ] + [∆ ] − ] E Z ( A )) = D ∆ , ∆ , ∆ , ∆ + ∆ + ∆ , ∆ + ∆ + ∆ + ∆ , ∆ + ∆ + ∆ + ∆ + ∆ E B ( A )) = D ∆ , ∆ + ∆ , ∆ + ∆ + ∆ , ∆ + ∆ + ∆ + ∆ E H ( A )) = D [∆ ] , [∆ ] + [∆ ] + [∆ ] + [∆ ] + [∆ ] E Z ( A ) = D ∆ , ∆ , ∆ , ∆ + ∆ + ∆ , ∆ + ∆ + ∆ + ∆ , ∆ + 2∆ + ∆ + 3∆ + ∆ + ∆ + ∆ E B ( A ) = D ∆ , ∆ + ∆ , ∆ + ∆ + ∆ , ∆ + ∆ + ∆ + ∆ + ∆ E H ( A ) = D [∆ ] , [∆ ] + 2[∆ ] + [∆ ] + 3[∆ ] + [∆ ] + [∆ ] + [∆ ] E Remark 6.
Extensions of the algebras A , A ( α ) α =1 , A , A ( α ) and A ( α, β ) α = ( β ± √ − β − β ) give algebras with -dimensional annihilator. Then, in the fol-lowing subsections we study the central extensions of the other algebras. Central extensions of A (1) . Let us use the following notations: ∇ = [∆ ] − [∆ ] , ∇ = [∆ ] + [∆ ] − [∆ ] , ∇ = [∆ ] − [∆ ] − [∆ ] + [∆ ] . The automorphism group of A consists of invertible matrices of the form φ = x y x z xy x t xy x w − y − xz − x y x . Since φ T α α α − α α − α − α − α α φ = α ∗∗∗∗ α ∗∗∗ α ∗ + α ∗ α ∗ α ∗ α ∗∗ α ∗ − α ∗ α ∗ + α ∗ − α ∗ − α ∗ − α ∗ α ∗ we have that the action of Aut( A ) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α , α ∗ = x α , α ∗ = x α . We have the following case:(1) If α = 0 , then choosing x = α α we have the representative h α ∇ + ∇ + ∇ i ; (2) If α = 0 , we have two representatives h∇ i and h∇ + ∇ i depending on whether α = 0 ornot.Consequently, we have the following algebras from A (1) : A ( α ) : e e = e e e = e e e = − e + αe e e = e e e = e e e = e + e e e = − e e e = − e e e = − e + e e e = − e e e = − ( α + 1) e e e = e A : e e = e e e = e e e = − e + e e e = e e e = e + e e e = − e e e = − e e e = − e e e = − e e e = − e e e = e A : e e = e e e = e e e = − e e e = e e e = e + e e e = − e e e = − e e e = − e e e = − e e e = e Central extensions of A ( α, β ) . Here we will consider the special cases for α = ( β ± p − β − β ) . The automorphism group of A ( α, β ) consists of invertible matrices of the form φ = x yx x z y x t y x v x ((2 − β + α ) z +(1+ α ) t )+ y x ( α − β + 4) xy ( α + β + 1) xy x . Let use the following notations: ∇ = [∆ ] − [∆ ] − [∆ ] , ∇ = [∆ ] + [∆ ] , ∇ = (2 β − ] + (2 αβ − β + 1)[∆ ] + ( α + 2 β − β + 1)[∆ ]+(3 α − αβ + 2 β − β + 1)[∆ ] + (2 α − αβ + 2 β − ] + (2 α − β + 1)[∆ ] . So, φ T α α (2 β − α αβ − β + 1) α ( α + 2 β − β + 1) α − α (3 α − αβ + 2 β − β + 1) α − α + α (2 α − αβ + 2 β − α α − β + 1) α φ == α ∗∗∗∗ α ∗∗∗ αα ∗ + α ∗ βα ∗ + α ∗ (2 β − α ∗ α ∗∗ α ∗ (2 αβ − β + 1) α ∗ ( α + 2 β − β + 1) α ∗ − β ) α ∗ − α ∗ (3 α − αβ + 2 β − β + 1) α ∗ − β + α ) α ∗ − α ∗ + + α ∗ (2 α − αβ + 2 β − α ∗ α − β + 1) α ∗ we have that the action of Aut( A ( α, β )) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α − β ( β − β − α − α x y,α ∗ = x α − ( β ( β − β −
1) + α (2 β − β + 3)) α x y,α ∗ = x α . We are interested only in the cases with α = 0 . Now we obtain the following cases:(1) For β ( β − β −
1) + α (2 β − β + 3) = 0 :(a) If β ( β − β − α − α = α ( β ( β − β −
1) + α (2 β − β + 3)) , then bychoosing x = √ α and y = α x β ( β − β − α (2 β − β +3) , we have the representative h∇ i ; (b) If β ( β − β − α − α = α ( β ( β − β −
1) + α (2 β − β + 3)) , then by choosing x = α ( α (2 β − β +3)+ β ( β − β − ) +2 α β ( β − α − β +1) β ( β − β − α (2 β − β +3) and y = α x β ( β − β − α (2 β − β +3) , and we have the representative h∇ + ∇ i . From the above cases we have new parametric algebras: A i ( β ) : e e = e e e = e e e = αe e e = βe e e = (2 β − e e e = e e e = e e e = (2 αβ − β + 1) e e e = ( α + 2 β − β + 1) e e e = (2 − β ) e e e = (3 α − αβ + 2 β − β + 1) e e e = ( α − β + 1) e e e = (2 α − αβ + 2 β − e e e = (2 α − β + 1) e A i +1 ( β ) : e e = e e e = e e e = αe e e = βe + e e e = (2 β − e e e = e e e = e e e = (2 αβ − β + 1) e e e = ( α + 2 β − β + 1) e e e = (2 − β ) e − e e e = (3 α − αβ + 2 β − β + 1) e e e = ( α − β + 1) e − e e e = (2 α − αβ + 2 β − e e e = (2 α − β + 1) e where i = 08 for α = ( β + p − β − β ) with β
6∈ { , } and where i = 10 for α = ( β − p − β − β ) with β = . (2) The condition β = 1 , for α = ( β + p − β − β ) gives α = 1 , that is A (1 , . Thebase of the second cohomology of this algebra spanned by elements: ∇ = [∆ ] − [∆ ] − [∆ ] , ∇ = [∆ ] + [∆ ] , ∇ = [∆ ] + [∆ ] + [∆ ] + [∆ ] + [∆ ] + [∆ ] . Since φ T α α α α α − α α α − α α α φ = α ∗∗∗∗ α ∗∗ α ∗ + α ∗ α ∗ + α ∗ α ∗ α ∗∗∗ α ∗ α ∗ α ∗ α ∗ − α ∗ α ∗ α ∗ − α ∗ + α ∗ α ∗ α ∗ we have that the action of Aut( A (1 , on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α , α ∗ = x α , α ∗ = α x . We are interested only in α = 0 , then we have the following cases:(a) If α = 0 , then for x = α α , α = α α we have the representative h α ∇ + ∇ + ∇ i . (b) If α = 0 , then also we have two cases:(i) If α = 0 , then x = α α , and we have the representative h∇ + ∇ i ; (ii) If α = 0 , then x = √ α , and we have the representative h∇ i ; Consequently, we have the following algebras from A (1 ,
1) : A (1) , A (1) and A ( α ) : e e = e e e = e e e = e + e e e = e + αe e e = e e e = e e e = e e e = e e e = e e e = e − αe e e = e e e = e + (1 − α ) e e e = e e e = e (3) The condition β = gives α = 1 for α = ( β + p − β − β ) , that is A (1 , ) . So,the second cohomology space of A (1 , ) spanned by elements: ∇ = [∆ ] − [∆ ] − [∆ ] , ∇ = [∆ ] + [∆ ] , ∇ = 2[∆ ] + [∆ ] + 2[∆ ] + [∆ ] + [∆ ] . Since φ T α α α α α − α α α − α α φ = α ∗∗∗∗ α ∗∗∗ α ∗ + α ∗ α ∗ + 3 α ∗ α ∗ α ∗∗ α ∗ α ∗ α ∗ α ∗ − α ∗ α ∗ α ∗ − α ∗ + α ∗ α ∗ we have that the action of Aut( A (1 , )) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α + x yα , α ∗ = x α , α ∗ = x α . Since α = 0 , and choosing y = − x α α , we have the representatives h∇ i and h∇ + ∇ i , depending on whether α = 0 or not.We have the following new -dimensional algebras constructed from A (1 , ) : A ( ) and A : e e = e e e = e e e = e + e e e = e e e = 2 e e e = e e e = e e e = e e e = 2 e e e = e e e = e e e = e e e = e Central extensions of A (0 , ) . If β = for α = ( β − p − β − β ) gives α = 0 , thatis A (0 , ) . So, the second cohomology space of A (0 , ) spanned by elements: ∇ = [∆ ] − [∆ ] − [∆ ] , ∇ = [∆ ]+[∆ ] , ∇ = 2[∆ ] − ] − ] − ]+[∆ ] − ] . Since φ T α α α − α − α − α − α α − α α − α φ = α ∗∗∗∗ α ∗∗ α ∗ α ∗ + α ∗ α ∗ α ∗∗∗ α ∗ − α ∗ − α ∗ − α ∗ + 3 α ∗ − α ∗ α ∗ − α ∗ + α ∗ α ∗ − α ∗ we have that the action of Aut( A (0 , )) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α + x yα , α ∗ = x α + 3 x yα , α ∗ = x α . We are interested in α = 0 , then we have the following cases: (1) If α − α = 0 , then x = √ α and y = − xα α , we have the representative h∇ i ; (2) If α − α = 0 , then x = − α +2 α α , y = − xα α and we have the representative h∇ + ∇ i . We have the following new -dimensional algebras constructed from A (0 , ) : A : e e = e e e = e e e = e e e = 2 e e e = e e e = e e e = − e e e = − e e e = e e e = − e e e = e e e = e e e = − e A : e e = e e e = e e e = e e e = e e e = 2 e e e = e e e = e e e = − e e e = − e e e = e e e = − e e e = e + e e e = e e e = − e Central extensions of A . Let us use the following notations: ∇ = [∆ ] , ∇ = [∆ ] + [∆ ] + [∆ ] + [∆ ] + [∆ ] . The automorphism group of A consists of invertible matrices of the form φ = x y x z xy x v xz + y x y x w xv + 2 yz x z + 3 xy x y x . Since φ T α α α
00 0 α α α φ = α ∗∗∗∗ α ∗ α ∗∗ α ∗∗∗ α ∗ α ∗ + α ∗ α ∗∗ α ∗∗∗ α ∗ α ∗∗ α ∗∗∗ α ∗ α ∗∗∗ α ∗ α ∗ , we have that the action of Aut( A ) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = x α , α ∗ = x α . We suppose that α = 0 , otherwise obtained algebra gives an algebra with 2-dimensional annihila-tor. Therefore, consider the following cases:(1) If α = 0 , then x = √ α , we have the representative h∇ i ; (2) If α = 0 , then x = q α α , we have the representative h∇ + ∇ i . Hence, we have the following new algebras: A : e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e A : e e = e e e = e e e = e e e = e e e = e e e = e + e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e Central extensions of A . Let us use the following notations: ∇ = [∆ ] , ∇ = [∆ ] + 2[∆ ] + [∆ ] + 3[∆ ] + [∆ ] + [∆ ] + [∆ ] . The automorphism group of A consists of invertible matrices of the form φ i = ( − k x y ( − k x ( − k z x + ( − k y x t xy + ( − k ( x + 2 z ) ( − k x + 3 y ( − k x ( − k , where k ∈ { , } . Since φ Ti α α α α α α α α φ i = α ∗∗∗∗ α ∗ α ∗∗ α ∗∗∗ α ∗ α ∗ + α ∗ α ∗ + α ∗∗ α ∗∗∗ α ∗ α ∗ α ∗∗∗ α ∗ α ∗∗∗ α ∗ α ∗ , we have that the action of Aut( A ) on the subspace h P i =1 α i ∇ i i is given by h P i =1 α ∗ i ∇ i i , where α ∗ = ( − i α − xα , α ∗ = α . We have only one non-trivial orbit with the representative h∇ i , and get A : e e = e e e = e e e = e e e = e e e = e e e = e + e e e = e + 2 e e e = e e e = e e e = e + 3 e e e = e e e = e e e = e e e = e e e = e Classification theorem.
Summarizing results of the present section we have the following the-orem.
Theorem B.
Let A be a -dimensional complex one-generated nilpotent assosymmetric algebra, then A is isomorphic to an algebra from the following list. A : e e = e e e = e e e = e e e = e e e = e e e = − e e e = e − e A ( α ) : e e = e e e = e e e = e e e = αe e e = e e e = e e e = (2 − α ) e e e = e − ( α − e A ( α, β ) : e e = e e e = e e e = αe + βe e e = e e e = e e e = e e e = − e + 2 e e e = ( α − e + ( β + 1) e A : e e = e e e = e e e = e e e = e e e = e + e e e = e e e = e e e = e e e = e e e = e A ( α ) : e e = e e e = e e e = − e + αe e e = e e e = e e e = e + e e e = − e e e = − e e e = − e + e e e = − e e e = − ( α + 1) e e e = e A : e e = e e e = e e e = − e + e e e = e e e = e + e e e = − e e e = − e e e = − e e e = − e e e = − e e e = e A : e e = e e e = e e e = − e e e = e e e = e + e e e = − e e e = − e e e = − e e e = − e e e = e i = 08 for α = ( β + p − β − β ) and i = 10 for α = ( β − p − β − β ) with β = A i ( β ) : e e = e e e = e e e = αe e e = βe e e = (2 β − e e e = e e e = e e e = (2 αβ − β + 1) e e e = ( α + 2 β − β + 1) e e e = (2 − β ) e e e = (3 α − αβ + 2 β − β + 1) e e e = ( α − β + 1) e e e = (2 α − αβ + 2 β − e e e = (2 α − β + 1) e i = 09 for α = ( β + p − β − β ) with β = and i = 11 for α = ( β − p − β − β ) with β = A i +1 ( β ) : e e = e e e = e e e = αe e e = βe + e e e = (2 β − e e e = e e e = e e e = (2 αβ − β + 1) e e e = ( α + 2 β − β + 1) e e e = (2 − β ) e − e e e = (3 α − αβ + 2 β − β + 1) e e e = ( α − β + 1) e − e e e = (2 α − αβ + 2 β − e e e = (2 α − β + 1) e A ( α ) : e e = e e e = e e e = e + e e e = e + 2 αe e e = e e e = e e e = e + αe e e = e e e = e e e = e e e = e e e = e + (1 − α ) e e e = e e e = e A : e e = e e e = e e e = e + e e e = e e e = 2 e e e = e e e = e e e = e e e = 2 e e e = e e e = e e e = e e e = e A : e e = e e e = e e e = e e e = 2 e e e = e e e = e e e = − e e e = − e e e = e e e = − e e e = e e e = e e e = − e A : e e = e e e = e e e = e e e = e e e = 2 e e e = e e e = e e e = − e e e = − e e e = e e e = − e e e = e + e e e = e e e = − e A : e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e A : e e = e e e = e e e = e e e = e e e = e e e = e + e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e e e = e A : e e = e e e = e e e = e e e = e e e = e e e = e + e e e = e + 2 e e e = e e e = e e e = e + 3 e e e = e e e = e e e = e e e = e e e = e Note: A ( ) ∼ = A ( ) . 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