On Leibniz superalgebras which even part is sl_2
Kh.A.Khalkulova, A.Kh.Khudoyberdiyev
TL;DR
This paper classifies Leibniz superalgebras with an even part isomorphic to sl_2, focusing on cases where the odd part is an irreducible module, revealing existence only when the odd part has dimension two.
Contribution
It provides a complete description of Leibniz superalgebras with even part sl_2 and irreducible odd modules, identifying the specific dimension where nontrivial structures occur.
Findings
Existence of Leibniz superalgebras with odd part only when dim L_1=2
Classification of such superalgebras with irreducible modules
Explicit description of the algebraic structures involved
Abstract
This article deals with a Leibniz superalgebras whose even part is a simple Lie algebra . We describe all such Leibniz superalgebras when odd part is an irreducible Leibniz bi-module on . We show that there exist such Leibniz superalgebras with nontrivial odd part only in case of
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Rings, Modules, and Algebras
On Leibniz superalgebras which even part is
Kh.A. Khalkulova1, A.Kh. Khudoyberdiyev1,2
1 Institute of Mathematics Academy of Science of Uzbekistan, 81, Mirzo Ulug’bek street, 100125, Tashkent, Uzbekistan, [email protected]
2 National University of Uzbekistan, 4, University street, 100174, Tashkent, Uzbekistan, [email protected]
Abstract.
This article deals with a Leibniz superalgebra whose even part is a simple Lie algebra . We describe all such Leibniz superalgebras when odd part is an irreducible Leibniz bi-module on . We show that there exist such Leibniz superalgebra with nontrivial odd part only in case of
Key words and phrases:
Leibniz algebra, Leibniz superalgebra, simple Lie algebra, Leibniz representation.
1991 Mathematics Subject Classification:
17A32, 17A36, 17B30, 13D10
1. Introduction
Extensive investigations in Lie algebras theory have led to the appearance of more general algebraic objects – Mal’cev algebras, binary Lie algebras, Lie superalgebras, Leibniz algebras and others.
During many years the theory of Lie superalgebras has been actively studied by many mathematicians and physicists. A systematic exposition of basic of Lie superalgebras theory can be found in [9]. Many works have been devoted to the study of this topic, but unfortunately most of them do not deal with nilpotent Lie superalgebras. In works [6], [8] the problem of the description of some classes of nilpotent Lie superalgebras have been studied.
Leibniz algebras have been first introduced by Loday in [12] as a non-antisymmetric version of Lie algebras. Leibniz superalgebras are generalizations of Leibniz algebras and, on the other hand, they naturally generalize Lie superalgebras. In the description of Leibniz superalgebras structure the crucial task is to prove the existence of suitable basis (the so-called adapted basis) in which the multiplication of the superalgebra has the most convenient form.
In the work [8] the Lie superalgebras with maximal nilindex were classified. Such superalgebras are two-generated and its nilindex equal to (where and are dimensions of even and odd parts, respectively). In fact, there exists unique Lie superalgebra of maximal nilindex. This superalgebra is filiform Lie superalgebra (the characteristic sequence equal to ) and we mention about paper [6], where some crucial properties of filiform Lie superalgebras are given.
For nilpotent Leibniz superalgebras it turns to be comparatively easy and was solved in [1]. The distinctive property of such Leibniz superalgebras is that they are single-generated and have the nilindex . The next step – the description of Leibniz superalgebras with dimensions of even and odd parts, respectively equal to and , and with nilindex were classified by applying restrictions the invariant such called characteristic sequences in [2], [4], [5], [7]. Solvable and semi-simple Leibniz superalgerbas are not investigated at this time. The first step of this assertion is describe Leibniz superalgebras which even part is a semi-simple Lie algebra. Note that the odd part of the superalgebra can be considered as a representation of the even part. Representation or bimodule of a Leibniz algebra is defined in [13] as a -module with two actions – left and right, satisfying compatibility conditions. In [3] it is established that any simple finite-dimensional Leibniz representation is either symmetric, meaning the left and the right actions differ by sign, or antisymmetric, meaning the left action is zero. The classical Weyl’s theorem on complete reducibility that claims any finite-dimensional module over a semisimple Lie algebra is a direct sum of simple modules does not generalize even for the simple Leibniz algebras case.
A Lie algebra can be considered as a Leibniz algebra and one can consider Leibniz representation of a Lie algebra. In [14] the authors describe the indecomposable objects of the category of Leibniz representations of a Lie algebra and as an example, in case the Lie algebra is the indecomposable objects in that category can be described, whereas for () they claim that it is of wild type.
Our main focus in this work is to describe Leibniz superalgebras even part is isomorphic to the three dimensional simple Lie algebra If the multiplication of the odd part is zero, then Leibniz superalgebra is isomorpic to Leibniz algebra, i.e., superalgebra with trivial odd part. We show that there exist such Leibniz superalgebra with nontrivial odd part only in case of
Throughout this work we shall consider spaces, algebras and superalgebras over the field of complex numbers.
2. Preliminaries
In this section we give necessary definitions and preliminary results.
Definition 2.1**.**
An algebra over a field is called a Leibniz algebra if it is defined by the Leibniz identity
[TABLE]
In fact for Leibniz algebra the ideal coincides with the space spanned by squares of elements of Moreover, it is readily to see that this ideal belongs to right annihilator, that is . Note that the ideal is the minimal ideal with respect to the property that the quotient algebra is a Lie algebra.
Definition 2.2**.**
Let be a Leibniz algebra, be a vector space and bilinear maps and satisfy the following three axioms:
[TABLE]
Then is called a representation of the Leibniz algebra or an bimodule.
Definition 2.3**.**
A -graded vector space is called a Leibniz superalgebra if it is equipped with a product which satisfies the following conditions:
[TABLE]
for all
The vector spaces and are said to be the even and odd parts of the superalgebra , respectively. It is obvious that is a Leibniz algebra and is a representation of Note that if in Leibniz superalgebra the identity
[TABLE]
holds for any and then the Leibniz superidentity can be transformed into the Jacobi superidentity. Thus, Leibniz superalgebras are a generalization of Lie superalgebras and Leibniz algebras.
Definition 2.4**.**
The set
[TABLE]
is called the right annihilator of a superalgebra
Using the Leibniz superidentity it is easy to see that is an ideal of the superalgebra . Moreover, the elements of the form () belong to .
Definition 2.5**.**
An -bimodule is called simple or irreducible, if it does not admit non-trivial -subbimodules. An -bimodule is called indecomposable, if it is not a direct sum of its -subbimodules. An -bimodule M is called completely reducible if for any -subbimodule there exists a complementing -subbimodule such that
In [3] it is proved that a finite-dimensional simple -bimodule is either symmetric or antisymmetric for any finite-dimensional Leibniz algebra .
Lemma 2.6**.**
[3]** Let be a finite-dimensional Leibniz algebra, and let be a finite-dimensional simple -bimodule. Then either or for all and
An -bimodule with trivial left actions is called symmetric. If the left action is the negative of the right action, then it is called antisymmetric.
In particular, in case of is isomorphic to three dimensional simple Lie algebra we have that there exist a basis of such that one of the table of multiplication is hold:
[TABLE]
[TABLE]
In [10] it is studied indecomposable complex finite-dimensional Leibniz algebra bimodule over that as a Lie algebra module is split into a direct sum of two simple -modules and it is proved that in this case there are only two indecomposable Leibniz -bimodules.
Theorem 2.7**.**
[10]**. An -module , where and are simple -modules is indecomposable as a Leibniz -bimodule if and only if Moreover, upto -bimodule isomorphism there are only two indecomposable -bimodules, which in basis have the following brackets:
[TABLE]
[TABLE]
Following theorem generalize of Theorem 2.7 in case of is a direct sum of simple -modules.
Theorem 2.8**.**
[11]** Let be an -bimodule and as a right -module let it decompose as where are simple -modules with base and Then is an indecomposable Leibniz -bimodule only if for all Moreover, up to -bimodule isomorphism there are exactly two indecomposable -bimodules:
[TABLE]
for all
[TABLE]
for all , where
3. Main part
In this section we describe Leibniz superalgebra which even part is .
Lemma 3.1**.**
Let be a Leibniz superalgebra, such that is a semi-simple Lie algebra. Then for any
Proof.
Note that for any an element belongs to the right annihilator of Consequantly, belongs to the center of the semi-simple Lie algebra Since the center of semi-simple Lie algebra is zero we have that ∎
Lemma 3.2**.**
Let be a Leibniz superalgebra, such that is a semi-simple Lie algebra and , then
Proof.
By the condition of the Lemma we have that for any and Considering the Leibniz superidentity for the elements and we have
[TABLE]
which follows that Since we derive that for any ∎
Let be a Leibniz superalgebra, i.e., even part is isomorphic to If is a simple -bimodule, then by Lemma 2.6 we have that either or for all
In case of , from Lemma 3.2, we derive that and is isomorphic to the Leibniz algebra with the following muliplication [15]:
[TABLE]
Therefore, it is sufficient to consider the case when for all In the following Proposition we describe such Leibniz superalgebras in case of
Proposition 3.3**.**
Let be a Leibniz superalgebra, such that is a simple bimodule. Let and for all then is isomorphic one of the following two Leibniz superalgebras:
[TABLE]
[TABLE]
Proof.
From Lemma 2.6 and (2.2) we have the following products:
[TABLE]
Put
[TABLE]
Consider following Leibniz superidentities:
- •
On the other hand: which implies
[TABLE]
- •
On the other hand: which derive
[TABLE]
- •
On the other hand: which implies
[TABLE]
- •
On the other hand: which derive
[TABLE]
The rest Leibniz superidentities give us the same restrictions. Thus, we have remaining multiplications:
[TABLE]
In case of we have superalgebra and if , then taking the change
[TABLE]
we obtain the Leibniz superalgebra . ∎
It should be Remark that in the superalgebra the multiplication is zero, thus is a Leibniz algebra.
In the following Proposition we investigate the case when
Proposition 3.4**.**
Let be a Leibniz superalgebra, such that is a simple bimodule. Let and for all then .
Proof.
By the condition of Proposition we have that there exist a basis of such that
[TABLE]
Put
[TABLE]
Now we consider Leibniz superidentity for the elements and
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
where
On the other hand,
[TABLE]
[TABLE]
[TABLE]
Comparing the coefficients at the basis elements, we obtain following restrictions:
[TABLE]
[TABLE]
[TABLE]
It is obvious that from (3.1) we have:
[TABLE]
From the first equality of (3.2) we derive:
[TABLE]
If is even, then we have On the other hand, Thus, we get which implies
If is odd, then in case of from the Leibniz superidentity we get If then considering Leibniz superidentity we obtain According to the equality (3.5) it follows
Since then from the equality (3.3) we have:
[TABLE]
Therefore, we get ∎
Now we consider the Leibniz superalgebra , such that is a split module into direct sum of the two simple -modules. Then by the Theorem 2.7 we obtain that as a -bimodule isomorphic to or
Proposition 3.5**.**
Let ba a Leibniz superalgebra such that is a bimodule isomorphic to . Then
Proof.
By the condition of the Proposition we have that there exist a basis of the superalgebra such that the following products are hold:
[TABLE]
Since
[TABLE]
we have that for Thus, we have for any values and
Put
[TABLE]
Considering Leibniz superidentities for the for the elements and we obtain following restrictions:
[TABLE]
[TABLE]
[TABLE]
Analogously, to the proof of Proposition 3.4 from (3.6) – (3.8) we conclude that Therefore, for any which implies
∎
Proposition 3.6**.**
Let ba a Leibniz superalgebra such that is a bimodule isomorphic to . Then
Proof.
By the condition of the Proposition we have that there exist a basis of the superalgebra such that the following products are hold:
[TABLE]
Since
[TABLE]
we have that for Thus, we have for any values and
Put
[TABLE]
Analogously, to the proof of Proposition 3.4 considering Leibniz superidentities for the elements and we obtain following restrictions:
[TABLE]
[TABLE]
[TABLE]
From the equations (3.9) – (3.11), similarly of the proof of Proposition 3.4 we conclude that Therefore, we obtain ∎
In the next two Propositions we consider Leibniz superalgebra , such that is a split modules as . By Theorem 2.8 we obtain that is a bimodule isomorphic to or .
Proposition 3.7**.**
Let ba a Leibniz superalgebra such that is a bimodule isomorphic to . Then
Proof.
By the condition of the Proposition we have that there exist a basis of the superalgebra such that the following products are holds:
[TABLE]
Since
[TABLE]
we have that 1\leq p\leq\big{[}\frac{k+1}{2}\big{]}, Then according to Lemma 3.1 we have
[TABLE]
Put
[TABLE]
Considering Leibniz superidentities for the elements and we have Which implies
∎
Proposition 3.8**.**
Let ba a Leibniz superalgebra such that is a bimodule isomorphic to . Then
Proof.
The proof of this Proposition is proved similarly to the previous one. ∎
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