The variety of nilpotent Tortkara algebras Текст научной статьи по специальности «Физика»

УДК 512.554.1

The variety of nilpotent Tortkara algebras

Ilya B. Gorshkov*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Ivan Kaygorodov^

CMCC, Universidade Federal do ABC Santo Andre, Brasil

Alexey A. Kytmanov*

Institute of Space and Information Technologies Siberian Federal University Kirensky, 26, Krasnoyarsk, 660074

Russia

Mohamed A. Salim§

United Arab Emirates University Al Ain, UAE

Received 06.11.2018, received in revised form 06.12.2018, accepted 06.08.2018 We give algebraic and geometric classification of all 5-dimensional nilpotent Tortkara algebras over complex field.

Keywords: Tortkara algebra, algeraic classification, central extension, geometric classification, degenerations.

DOI: 10.17516/1997-1397-2019-12-2-173-184.

The algebraic classification (up to isomorphism) of algebras of dimension n from a certain variety defined by some family of polynomial identities is a classic problem in the theory of non-associative algebras. There are many results related to algebraic classification of small dimensional algebras in varieties of Jordan, Lie, Leibniz, Zinbiel and many other algebras [16-19, 25,28,37]. Another interesting direction in classifications of algebras is a geometric classification. There are many results related to geometric classification of Jordan, Lie, Leibniz, Zinbiel and many other algebras [3-6,8,9,11,12,29-33,35-38,41]. In the present paper, we give algebraic and geometric classifications of nilpotent algebras of a new class of non-associative algebras introduced by Dzhumadildaev in [21].

Zinbiel algebras were introduced by Loday in [39]. They were studied in [15,20,22,23,35,40, 44]. Under the Koszul duality, the operad of Zinbiel algebras is dual to the operad of Leibniz algebras. Zinbiel algebras are related to Tortkara algebras [21] and Tortkara triple systems [10]. An algebra A is called a Zinbiel algebra if it satisfies the identity

(xy)z = x(yz + zy).

Every Zinbiel algebra with the commutator multiplication defines a Tortkara algebra. An anti-commutative algebra A is called a Tortkara algebra if it satisfies the identity

(ab)(cb) = J(a, b, c)b, where J(a, b, c) = (ab)c + (bc)a + (ca)b.

* ilygor8@gmail.com

1 kaygorodov.ivan@gmail.com

^ aakytm@gmail.com

§ msalim@uaeu.ac.ae © Siberian Federal University. All rights reserved

It is easy to see that every metabelian Lie algebra (i.e., (xy)(zt) = 0) is a Tortkara algebra and every Dual Mock-Lie algebra (i.e., antiassociative and anticommutative) is a Tortkara algebra.

Our method of classification of nilpotent Tortkara algebras is based on calculation of central extensions of smaller nilpotent algebras from the same variety. Central extensions play an important role in quantum mechanics: one of the earlier encounters is by means of Wigner's theorem which states that symmetry of a quantum mechanical system determines an (anti-)unitary transformation of a Hilbert space. Another area of physics where one encounters central extensions is the quantum theory of conserved currents of a Lagrangian. These currents span an algebra which is closely related to so called affine Kac-Moody algebras, which are the universal central extension of loop algebras. Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way, the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac-Moody algebras have been conjectured to be a symmetry groups of a unified superstring theory. The centrally extended Lie algebras play a dominant role in quantum field theory, particularly in conformal field theory, string theory and in M-theory. In the theory of Lie groups, Lie algebras and their representations, a Lie algebra extension is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is a trivial extension obtained by taking a direct sum of two Lie algebras. Other types are split extension and central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. A central extension and an extension by a derivation of a polynomial loop algebra over finite-dimensional simple Lie algebra give a Lie algebra which is isomorphic to a non-twisted affine Kac-Moody algebra [7, Chapter 19]. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra, the Heisenberg algebra is the central extension of a commutative Lie algebra [7, Chapter 18]. The algebraic study of central extensions of Lie and non-Lie algebras has a very rich history [2,26,27,34,42,43,45]. In particular, Skjelbred and Sund used central extensions of Lie algebras for a classification of nilpotent Lie algebras [42]. Following that, there were described all non-Lie central extensions of all 4-dimensional Malcev algebras [27], all non-associative central extensions of 3-dimensional Jordan algebras [26], all anticommutative central extensions of 3-dimensional anticommutative algebras [13], and all central extensions of 2-dimensional algebras [14] by means of the method described by Skjelbred and Sund. We also note that the method of central extensions is an important tool in the classification of nilpotent algebras (see, e.g., [24]). Using this method, there were described all 4-dimensional nilpotent associative algebras [18], all 5-dimensional nilpotent Jordan algebras [25], all 5-dimensional nilpotent restricted Lie agebras [17], all 6-dimensional nilpotent Lie algebras [16,19], all 6-dimensional nilpotent Malcev algebras [28], and some others.

1. The algebraic classification of 5-dimensional nilpotent Tortkara algebras

1.1. Preliminaries

A method for classification of nilpotent algebras

Throughout this paper, we use the notations and methods introduced in [14, 26, 27] and adapted for the Tortkara case with some modifications. Below we give some of the most important definitions.

Let (A, •) be a Tortkara algebra over an arbitrary base field k of characteristic not 2 and V a vector space over the same base field k. Then the k-linear space Z2 (A, V) is defined as the

set of all skew-symmetric bilinear maps 9 : A x A —> V such that

9(xy, zt) + 9(xt, zy) = 9( J(x, y, z),t) + 9( J(x, t, z), y).

Its elements are called cocycles. For a linear map f: A —> V, we define Sf: A x A —> V by Sf (x,y) = f (xy) so that Sf e Z2 (A, V). Define B2 (A, V) = {9 = Sf: f e Hom (A, V)}. One can easily check that B2(A, V) is a linear subspace of Z2 (A, V) whose elements are called coboundaries. We define the second cohomology space H2 (A, V) as the quotient space Z2 (A, V) /B2 (A, V).

Let Aut (A) be the automorphism group of the Tortkara algebra A and let $ e Aut(A). For 9 e Z2 (A, V) define $9 (x, y) = 9 ($ (x), $ (y)). Then $9 e Z2 (A, V). So, Aut (A) acts on Z2 (A, V). It is easy to verify that B2 (A, V) is invariant under the action of Aut (A) and so we have that Aut (A) acts on H2 (A, V).

Let A be a Tortkara algebra of dimension m < n over an arbitrary base field k of characteristic not 2, and V be a k-vector space of dimension n — m. For any 9 e Z2 (A, V), define the bilinear product " [—, — ]Ae" on the linear space Ag := A © V by [x + x', y + y']Ae = xy + 9 (x, y) for all x,y e A, x',y' e V. The algebra Ag is a Tortkara algebra which is called an (n — m)-dimensional central extension of A by V. Indeed, clearly, Ag is a Tortkara algebra if and only if 9 e Z2 (A, V).

We also call the set Ann(9) = {x e A : 9 (x, A) = 0} the annihilator of 9. Recall that the annihilator of an algebra A is defined as the ideal Ann (A) = {x e A : xA = 0}, and observe that Ann (Ag) = Ann(9) n Ann (A) © V.

We now state the following key result:

Lemma 1.1. Let A be an n-dimensional Tortkara algebra with dim(Ann(A)) = m = 0. Then, there exists (up to an isomorphism) a unique (n — m)-dimensional Tortkara algebra A' and a bilinear map 9 e Z2 (A, V) with Ann(A) n Ann(9) = 0, where V is a vector space of dimension m such that A = Ag and A/ Ann(A) = A '.

Proof. Let A' be a linear complement of Ann(A) in A. Define a linear map P : A —> A' by P(x + v) = x for x e A' and v e Ann(A) and define a multiplication on A' by [x, y]A' = P(xy) for x, y e A . Then, for x, y e A,

P(xy) = P((x — P(x) + P(x))(y — P(y) — P(y))) = P(P(x)P(y)) = [P(x), P(y)]A'.

Since P is a homomorphism, P(A) = A' is a Tortkara algebra and A/ Ann(A) = A', which gives us the uniqueness. Now, define the map 9 : A' x A' —> Ann(A) by 9(x, y) = xy — [x, y]A/. Thus, Ag is A and therefore 9 e Z2(A, V) and Ann(A) n Ann(9) = 0. The lemma is proved. □

However, in order to solve the isomorphism problem, we need to study the action of Aut (A) on H2 (A, V). To do that, fix e1,..., es, a basis of V, and 9 e Z2 (A, V). Then, there is a unique

s

representation 9 (x,y) = Y19% (x,y) ei with 9i e Z2 (A, k). Moreover, Ann(9) = Ann(9i) n

i=i

Ann(92) n • • • n Ann(9s). Furthermore, 9 e B2 (A, V) if and only if all 9% e B2 (A, k).

Given a Tortkara algebra A, if A = I © kx is a direct sum of two ideals, then kx is called an annihilator component of A.

Definition 1. A central extension of an algebra A without annihilator component is called a non-split central extension.

It is not difficult to show (see [27, Lemma 13]), that given a Tortkara algebra Ag, if use the

s

respresentation 9 (x,y) =^2,9% (x,y) e Z2 (A, V), and Ann(9) n Ann (A) = 0, then Ag has an

i=i

annihilator component if and only if [91], [92],..., [9s] are linearly dependent in H2 (A, k).

Let V be a finite-dimensional vector space over k. The Grassmannian Gk (V) is a set of all fc-dimensional linear subspaces of V. Let Gs (H2 (A, k)) be the Grassmannian of subspaces of dimension s in H2 (A, k). There is a natural action of Aut (A) on Gs (H2 (A, k)). Let $ £ Aut (A). For W = (N , fa], • • •, [0s]> £ Gs (H2 (A, k)) define = ([$01], , • • •, [№s]). Then £ Gs H2 (A, k)). Denote the orbit of W £ Gs (H2 (A, k)) under the action of Aut (A) by Orb (W). Since given

Wi = ([01], [02], • • •, [0S]>, W2 = ([$1], [$2], • • •, [$s]> £ Gs (H2 (A, k),)

s s

we easily obtain that if W1 = W2, then n Ann(0¿) n Ann (A) = n Ann($^ n Ann (A) so that

i=1 i=1

we can introduce the set

Ts (A) = {W = ([01 ], [02], • • •, [0s]> £ Gs (H2 (A, k)) : Jn 1 Ann(0i) n Ann (A) = 0} ,

which is stable under the action of Aut (A).

Now, let V be a s-dimensional linear space. By E (A, V) we denote the set of all non-split s-dimensional central extensions of A by V. Then,

E (A, V) = j Ag : 0 (x, y) = ¿0i (x, y) eiand ([01], [02], • • •, [0s]> £ Ts (A) j •

We now state the following result which can be proved as [27, Lemma 17].

s

Lemma 1.2. Let Ag, A$ £ E (A, V). Suppose that 0 (x,y) = 0i (x,y) ei and $ (x,y) =

i=1

s

= Y1 (x, y) ei. Then the Tortkara algebras Ag and A$ are isomorphic if and only if

i=1

Orb ([01] , [02] , • • • , [0s]> = Orb ([$1] , [$2] , • • • , [$s]> •

Consequently, there exists a one-to-one correspondence between the set of Aut (A)-orbits on Ts (A) and the set of isomorphism classes of E (A, V). We therefore have the following procedure that allows us, given the Tortkara algebras A of dimension n, to construct all non-split central extensions of A •

Procedure

1. For a given Tortkara algebra A of dimension n, determine H2(A , k) and Aut(A ).

2. Determine the set of Aut(A )-orbits on Ts (A ).

3. For each orbit, construct the non-split Tortkara algebra corresponding to its representative. Notations

We now introduce some notations. Let A be a Tortkara algebra with a basis e1,e2, • • • ,en. Then, by Aj we denote the Tortkara bilinear form Aij : A x A —> k with Aij- (e¡, em) =0 if {i,j} = {l, m} and Aij (ei, ej) = -Aij (e¿, ei) = 1. Then, the set {Aij : 1 < i < j < n} is a basis for the linear space of bilinear forms on A. Then, every 0 £ Z2 (A, V) can be uniquely written as 0 = cij Aij, where cij £ k. We use the following notations as well:

Tj — jth i-dimensional Tortkara algebra, Ni — i-dimensional algebra with zero product, (A)i,j — jth i-dimensional central extension of A^

Central extensions of nilpotent low dimensional Tortkara algebras

There are no nontrivial 1- and 2-dimensional nilpotent Tortkara algebras. There is only one nontrivial 3-dimensional nilpotent Tortkara algebra (it is the non-split central extention of N2, see T0! below). Thanks to [13], we have a description of all anticommutative central extensions of 3-dimensional anticommutative nilpotent (Tortkara) algebras:

N 1 (N2)3,i eie2 = e3,

TO2 (T3i)4,i eie2 = e3, eie3 = e4,

TO3 (N3)5,i eie2 = e4, eie3 = e5,

TO4 (T3i)5,i eie2 = e3, eie3 = e4, e2 e3 = e5

1.2. The algebraic classification of 5-dimensional nilpotent Tortkara algebras

The algebraic classification of 4-dimensional nilpotent Tortkara algebras

A multiplication tables H2 (A) Automorphisms

TOi eie2 = e3 [Ai3], [Ai4], [A23], [A24], [A34] (x y 0 0\ zu 0 0 v h xu — zy g U r 0 tj

TO2 eie2 = e3, eie3 = e4 [Ai4], [A23], [A24] x 0 0 0 y z 0 0 u v xz 0 h g xv x2 z

1-dimensional central extensions of N4

Thanks to [41], we have all non-split 1-dimensional anticommutative central extensions of N4 :

Tq5 : (N4)5,1 : eie2 = e5, €364 = e5.

1-dimenisonal central extensions of TO

01

Since

x y 0 0 t ( 0 0 ai a2 x y 0 0 / 0 a* * a* a32

z u 0 0 0 0 a3 a4 z u 0 0 ~a* 0 * a3 * a4

v h xu — zy g —ai —a3 0 a5 v h xu — zy g = * —a* * — a *3 0 * a35

l r 0 t \ —a2 —a4 —a5 0 l r 0 t \ * ~ai * — a *4 * — a 53 0

where

a al

(aix + a^z — a.5¡)h + (-a^y — a4u)l + (a^x + a^z + a5v)r — v(aiy + a^u),

(aix + a3z — a5jl)(ux — yz),

(aix + a3z — a5l)g + (a2 x + a4z + a5v)t,

(aiy + a3u — a5 r)(ux — yz),

(aiy + a3u — a5 r)g + (a2y + a4u + a5h)t,

( ux — yz) a5 t,

a =

a

a

a=

we obtain that the action of Aut (T^) on a subspace

(ai[Ai3j + &2 [A14] + a3[A23] + a4[A24] + «5^34]}

is given by

(«1 [A 13] + o£[Ai4] + «3 [A23] + a4 [A24] + «5 [A34]}. We provide the orbit of every possible case:

1 Tf ^ A j-u U A i aix + «3z a2X + a.4Z aiy + «3«

1. If a5 = 0, then choosing g = 0, l =--, v =--, r =--,

«5

a5

a5

a2y + «4 u

h =--, we have a representative {(ux — yz)ta5[A34]}, and the orbit is ([A34]}.

a5

2. If a5 = 0, ai = a3 = 0, then we can assume that a2 =0 and a4 = 0, and, choosing

a2 y

u =--, we have a representative {(a2x + a4z)t[Ai4]}, and the orbit is ([Ai4]}.

a4

3. If a5 =0 and ai =0 or a3 = 0, we can assume that ai =0 and a3 = 0. Then

(a) if a2a3 = aia4, then choosing g = — — t and x = y + — (u — z) we have a represen-

a3 ai

tative {(aix + a3z)(ux — yz)([Ai3] + [A23])} and the orbit is ([Ai3] + [A23]}.

(b) if a2a3 = aia4, then we can suppose that a4 = 0. Choosing y

a^xi

a3(a2x + a4i)'

aix a2y

z =--, u =--we have a representative {(aiy + a3u)(ux — yz)([Ai4] + [A23])}

a3 a4

and the orbit is ([Ai4] + [A23]}. It is easy to verify that all the previous orbits are different so that we arrive at

Ti(T4i) = ([Ai3] + [A23]) u ([Ai4^ U ([Ai4] + [A23^ u ([A34^.

Note that the orbit ([Ai3] + [A23]} gives a split central extension and the orbit ([Ai4]} gives a non-split central extension with 2-dimensional annihilator (it is a 2-dimensional central extension of a 3-dimensional algebra). Now we have all non-split 1-dimensional central extensions with 1-dimensional annihilator of T0i :

T56 : (T4i)5,i : eie2 = e3, eie4 = e5, e2e3 = e5, T[;7 : (T0i)5i2 : eie2 = e3, e3e4 = e5.

1-dimensional central extensions of Ti

02

Since

/ 0 a* a** a*\ x 0 0 0 t / 0 0 0 a1 x 0 0 0

—a* 0 * a* * a3 y z 0 0 0 0 a2 a3 y z 0 0

—a** * — a *2 0 0 u v xz 0 0 —a2 0 0 u v xz 0

V * — a *i * — a *3 0 0 h g xv x2zj V —a1 —a3 0 0 h g xv x2 z

with

a* = (—a.2u — a3h)z + a3gy + aigx + a.2yv,

a** = (a2z + a3v)yx + aiv2,

al = (aix + a3y)x2z,

a2 = (a2z + a3v)xz, 22

a* = a3z2x2,

we obtain that the action of Aut (T|) on a subspace (a^A14] + a2[A23] + a3[A24]) is given by

« [A 14] + a} [A23]+ «3 [A24]). We provide the orbit of every possible case:

«2 we have the orbit is ([A24]).

1 ai

1. If «5=0, then choosing x =1,z =-,y =--,v = — - _

V«3 «3 V(a3)3

2. If a5 = 0, then we have one from the following orbits ([A14]), ([A14] + [A23]), ([A23]). It is easy to verify that all the previous orbits are different so that we arrive at

Ti(T02) = ([A14]) U ([A14] + [A23]) U ([A23]) U ([A24]).

We therefore have all non-split 1-dimensional central extensions with 1-dimensional annihi-lator of Tq2:

T08 (T02)5 ,1 eie2 = e3, eie3 = eo e1 e0 = e5,

T09 (T02)5 , 2 eie2 = e3, eie3 = eo e1 e0 = e5, e2e3

T10 (T02)5 ,3 eie2 = e3, eie3 = eo, e2 eo = e5.

The algebraic classification of 5-dimesional nilpotent Tortkara algebras

Theorem 1.1. Let A be a nontrivial 5-dimensional nilpotent Tortkara algebra. Then, A is isomorphic to exactly one of the following algebras:

T0i e1e2 = e3,

TO2 e1e2 = e3, e1 e3 = eo,

T03 e1e2 = eo, e1 e3 = e5.

TOo e1e2 = e3, e1 e3 = eo, e2e3 = e5,

TO5 e1e2 = e5, e3eo = e5,

TOe e1e2 = e3, eieo = e5, e2e3 = e5,

TO7 e1e2 = e3, e3eo = e5,

T08 e1e2 = e3, e1 e3 = eo, e1 eo = e5,

TO9 e1e2 = e3, e1 e3 = eo, e1 eo = e5, e2e3

T10 e1e2 = e3, e1 e3 = eo, e2eo = e5.

2. The geometric classification of 5-dimesional nilpotent Tortkara algebras

2.1. Preliminaries

Definitions and notation

Given an n-dimensional vector space V, the set Hom(V ® V, V) = V3 ® V3 ® V is a vector space of dimension n3. This space has a structure of the affine variety Cn . Indeed, fix a basis e1,... ,en of V. Then, any / G Hom(V ® V, V) is determined by n3 structure constants ck j G C

n

such that /(ei ® ej) = ^ ck jek. A subset of Hom(V® V, V) is Zariski-closed if it can be defined

k = 1

by a set of polynomial equations in the variables ckj (1 < i,j, k < n).

Let T be a set of polynomial identities. All algebra structures on V satisfying polynomial identities from T form a Zariski-closed subset of the variety Hom(V ® V, V). We denote this subset by L(T). The general linear group GL(V) acts on L(T) by conjugations:

(g * ^)(x ® y) = g^(g-1x ® g-iy)

for xy e V, n e L(T) C Hom(V ® V, V) and g e GL(V). Thus, L(T) is decomposed into GL(V)-orbits that correspond to isomorphism classes of algebras. Let O(p) denote the orbit of H G L(T) under the action of GL(V) and O(p) denote the Zariski closure of O(^).

Let A and B be two n-dimensional algebras satisfying identities from T and n,A e L(T) represent A and B, respectively. We say that A degenerates to B and denote A fi B if A e O(^). Note that, in this case, we have O(A) C O(^). Hence, the definition of a degeneration does not depend on the choice of n and A. If A ^ B, then the assertion A fi B is called a proper degeneration. We write A fi B if AG O(^).

Let A be represented by n e L(T). Then A is rigid in L(T) if O(p) is an open subset of L(T). Recall that a subset of a variety is called irreducible if it cannot be represented as a union of two non-trivial closed subsets. A maximal irreducible closed subset of a variety is called an irreducible component. It is well known that any affine variety can be represented as a finite union of its irreducible components in a unique way. The algebra A is rigid in L(T) if and only if O(p) is an irreducible component of L(T).

Given spaces U and W, we write simply U > W instead of dim U > dim W.

A method for description of all degenerations of algebras

In the present work we use the methods from [12,29,30,41] applied to Lie algebras. First of all, if A —y B and A = B, then Der(A) < Der(B), where Der(A) is the Lie algebra of derivations of A. We will compute the dimensions of algebras of derivations and will check the assertion A fi B only for such A and B that Der(A) < Der(B). Secondly, if A fi C and C fi B then A fi B. If there is no C such that A fi C and C fi B are proper degenerations, then the assertion A fi B is called a primary degeneration. If Der(A) < Der(B) and there are no C and D such that C fi A, B fi D, C fi D and one of the assertions C fi A and B fi D is a proper degeneration, then the assertion A fi B is called a primary non-degeneration. It suffices to prove only primary degenerations and non-degenerations to describe degenerations in the variety under consideration. It is easy to see that any algebra degenerates to the algebra with zero multiplication. From now on, we use this fact without mentioning it.

To prove primary degenerations, we construct families of matrices parametrized by t. Namely, let A and B be two algebras represented by the structures n and A from L(T), respectively. Let ei,... ,en be a basis of V and ckj (1 < i,j, k < n) be the structure constants of A in this basis. If

n

there exist aj(t) e C (1 < i,j < n, t e C4) such that Ej = J2 aj. (t)ej (1 < i < n) form a basis of

j=i

V for any t e C4, and the structure constants of n in the basis E\,... ,Efn are such polynomials ckj(t) e C[t] that ckj (0) = ckj, then A fi B. In this case, E[,... ,Efn is called a parametrized basis for A fi B.

2.2.The geometric classification of 5-dimensional nilpotent Tortkara algebras

The main result of the present section is the following theorem.

Theorem 2.1. The variety of 5-dimensional nilpotent Tortkara algebras has one irreducible component defined by the rigid algebra T50. The graph of primary degenerations for 5-dimensional nilpotent Tortkara algebras has the following form:

Proof. Note that the set {T^ji^^g U {N5} gives the variety of all 5-dimensional nilpotent Malcev algebras. The descrition of all degenerations of 5-dimensional nilpotent Malcev algebras was obtained in [36]. By calculation of the dimension of the algebra of derivations of T50, we have the dimension of the orbit closure of Ti0.

The parametrized basis formed by 1 1

1

E\ = tei + e2, E2, = te2 + 7e3, El3 = t2e3 + e4, E\ = tóe4 + ^e5, E5 = te5

t2

gives the degeneration T50 ^ T59. The parametrized basis formed by

E{ = e1, E2 = e3, E3 = e4, E4 = te2, Ef5 = -te5 gives the degeneration T50 ^ T07. Thus, the theorem is proved.

□

The first part of this work is supported by the Russian Science Foundation under grant 18-7110007. The second part of this work was supported by UAEU UPAR (9) 2017 Grant G00002599 (Mohamed A. Salim).

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