Constructions and Properties for a Finite-Dimensional Modular Lie Superalgebra

Abstract

In this paper, a finite-dimensional Lie superalgebra over a field of prime characteristic is constructed. Then, we study some properties of . Moreover, we prove that is an extension of a simple Lie superalgebra, and if , then it is isomorphic to a subalgebra of a restricted Lie superalgebra.

1. Introduction

In the 1970s, physicists introduced the concept of Lie superalgebra in order to describe supersymmetry (see [1]). Since Lie superalgebra is an important mathematical model of supersymmetry, the research on it has been very active and rich results have been obtained (see [2]). In mathematics, Lie superalgebra is also a natural generalization of Lie algebra. In 1977, Kac completed the classification of finite-dimensional simple Lie superalgebras over a field of characteristic zero (see [3]). The research on Lie superalgebras over a field of characteristic zero has been quite systematic (see [4–6]), but the research on modular Lie superalgebras remains to be perfected (see [7]). Although some mathematicians try to study the classification of modular Lie superalgebras (see [7–13]), the classification problem has still been open. Therefore, it is very important to construct new finite-dimensional modular Lie superalgebras.

The finite-dimensional modular Lie superalgebras , were constructed in [14, 15], respectively. Their natural filtrations are investigated in [16]. The finite-dimensional modular Lie superalgebra was given in [17]. Modular Lie superalgebra , which takes the Grassmann algebra as base algebra, was constructed in [18]. Inspired by the above mentioned literatures, this paper constructs a finite-dimensional modular Lie superalgebra of contact type, which is denoted by .

The remainder of this paper is arranged as follows. A brief summary of the relevant concepts and notations is presented in Section 2. In Section 3, we construct the finite-dimensional modular Lie superalgebra . In Section 4, we obtain some properties of . Moreover, we prove that is an extension of , and if , then it is isomorphic to a subalgebra of .

2. Preliminaries

Throughout this paper, denotes an algebraic closed field of characteristic ; is an integer greater than 3. Apart from the standard notation , the sets of positive integers and nonnegative integers are denoted by and , respectively. denotes the ring of integers modulo 2.

Let be the Grassmann algebra over in variables . Set and , where . For , set , and . Then, is an -basis of .

Let be the Grassmann algebra over in variables . Obviously, .

Let be the linear map such that for any , ,where and .

Let . Then, is a finite-dimensional Lie superalgebra according to the operations in . Let . Then, , where . In [18], Xin proves that is not a simple Lie superalgebra.

Let be the tensor product, where is the truncated polynomial algebra satisfying for all (see [17]). Then, is an associative superalgebra with -gradation induced by the trivial -gradation of and the natural -gradation of . Namely, , where and .

For and , we abbreviate as . Then, the elements with and form an -basis of . It is easy to see that is a -graded superalgebra, where . In particular, and , where .

In this paper, let , where is a superalgebra. If is a -homogeneous element of , then denotes the -degree of .

Set . Given , let be the partial derivative on with respect to . For , let be the linear transformation on such that for all and . Let denote the derivation superalgebra of (see [11]). Then, for all since (see [7]).

Suppose that and . When , denotes the uniquely determined element of satisfying . Then, the number of integers less than in is denoted by . When , we set and . Therefore, for all and .

We define for and . Since the multiplication of is supercommutative, is a derivation of . Let

Then, is a finite-dimensional Lie superalgebra contained in . A direct computation shows thatwhere and .

Definition 1 (see [4]). A Lie superalgebra is called simple if it does not have any graded ideals which are different from and and if, moreover, .

3. Construction of

Set . Let be the linear map such thatwhere , , and .

Let  = . Then, is a subspace of .

Let

By direct calculation, we havewhere and is the Kronecker delta.

Proposition 1. .

Proof. For any and , we haveTherefore,

Proposition 2. Let and , where . Then,where .

Proof. Let .
Since , we getTherefore, , for .
Hence,Since , where , we obtainSet . For any , we haveIt follows thatFor all , we haveThen, exactly equals to the coefficient of in .
In addition,Then, exactly equals to the coefficient of in .
Therefore, .
An immediate corollary of this proposition is the following.

Corollary 1. is a subalgebra of .
Next, we give another way to express . We still denote the linear map from to by . Namely,

Then, we prove the following proposition.

Proposition 3. .

Proof. Let . We obtainTherefore, . Then, is injective.
We define an operator in . For any , we haveBy Proposition 2, we haveNote that is injective. Therefore, it can be concluded that is a Lie superalgebra about the operator (see [1]). It follows from equation (21) that as Lie superalgebras.
Since , we use instead of defined by equation (20). By equations (5) and (6) and Proposition 2, for , we have

Definition 2. .

Proposition 4. , where .

Proof. Let . It suffices to prove that . For any , let , where . If generator is contained in , then . Otherwise, .
Firstly, we prove :(i)If , there exist such that Therefore, .(ii)If , , then . Therefore,According to (i) and (ii), . Namely, for all , we have . Therefore, .
Secondly, we prove :
Note thatTherefore, it suffices to prove that . Without loss of generality, suppose that there exist such that , where .
If , thenIf , then , where .
Therefore, . Namely,By the definition of the Grassmann algebra, for all , we haveIt follows that . Then, .
Therefore, .

4. Some Properties of

Proposition 5. does not possess a -graded structure as .

Proof. Suppose that has -gradation:where . Let , where . Suppose that , for all . Since , we have . Since , we have . Then, . Therefore, for all . Since and , we obtain . For all , we have . Then, , where . Following the discussion above, we have , where .
On the other hand, let , where . Since , where , we know . Therefore, . For all , we have , where . Then, . Following the discussion above, we have , where .
If , then . If , then .
Therefore, does not possess a -graded structure as .

Lemma 1 (see [15]). Let . Then, is an ideal of and .

Theorem 1. Let . Then, is an ideal of . Namely, the Lie superalgebra is not simple.

Proof. Let , where . Then,where . Since is an ideal of , we have . Then, . Therefore, is an ideal of . Now, we conclude that is not a simple Lie superalgebra.
Let . Then,Let . Then, we define an operator in . For all , , we define . Then, is a simple Lie superalgebra.

Theorem 2. . Therefore, is an extension of the simple Lie superalgebra and is the maximal ideal of .

Proof. We define a linear map such that for all . Obviously, is an isomorphism of linear spaces.
In addition, for all , we haveTherefore, is a homomorphism of Lie superalgebras. Then, is an isomorphism of Lie superalgebras. We consider the following sequence:In the above sequence, is the embedded map from to and is the natural homomorphism from to . Obviously, . Therefore, the above sequence is an exact sequence. Note that is an ideal of and . Therefore, is an extension of the simple Lie superalgebra .
Let denote the divided power algebra over with basis . Let . For , let be the linear transformation of the superalgebra such thatLet . Then, we define a linear map such thatfor all . Let . Then, is a finite-dimensional odd contact Lie superalgebra (see [19]).

Proposition 6 (see [15]). Let . Then, is isomorphic to .

In [15], we can see that the isomorphism can be extended to an isomorphism .

Theorem 3. If , then is isomorphic to a subalgebra of .

Proof. Let . It is a subalgebra of . Then, we define a map such that for all . By virtue of Proposition 6, is an isomorphism of Lie superalgebras.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2572021BC02) and the National Natural Science Foundation of China (Grant No. 11626056).