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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 891241, 7 pages
The Natural Filtration of Finite Dimensional Modular Lie Superalgebras of Special Type
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Received 21 May 2013; Accepted 8 July 2013
Academic Editor: Shi Weichen
Copyright © 2013 Keli Zheng and Yongzheng Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with the natural filtration of Lie superalgebra of special type over a field of prime characteristic. We first construct the modular Lie superalgebra . Then we prove that the natural filtration of is invariant under its automorphisms.
Although many structural features of nonmodular Lie superalgebras (see [1–3]) are well understood, there seem to be very few general results on modular Lie superalgebras. The treatment of modular Lie superalgebras necessitates different techniques which are set forth in [4, 5]. In , four series of modular graded Lie superalgebras of Cartan type were constructed, which are analogous to the finite dimensional modular Lie algebras of Cartan type  or the four series of infinite dimensional Lie superalgebras of Cartan type defined by even differential forms over a field of characteristic zero . Recent works on the modular Lie superalgebras of Cartan type can also be found in [9–13] and references therein.
It is well known that filtration techniques are of great importance in the structure and the classification theories of Lie (super)algebras (see [1, 2, 14, 15]). For some classes of modular Lie (super)algebras, the filtrations have been well investigated, for example, the natural filtrations of finite dimensional modular Lie algebras of Cartan type [16, 17] and of finite dimensional simple modular Lie superalgebras , , and of Cartan type [18, 19].
The original motivation for this paper comes from the researches of structures for the finite dimensional modular Lie superalgebras and , which were first introduced in [20, 21], respectively. The starting point of our studies is to construct a class of finite dimensional modular Lie superalgebras of special type, which is denoted by . A brief summary of the relevant concepts and notations in the finite dimensional modular Lie superalgebras is presented in Section 2. In Section 3, by using the ad-nilpotent elements of , we show that the natural filtration of is invariant under its automorphisms.
Throughout this paper, denotes an algebraic closed field of characteristic , and is an integer greater than . In addition to the standard notation , we write and to denote the sets of positive integers and nonnegative integers, respectively.
Let be the Grassmann algebra over in variables . Set and , where . For , set , and . Then is an -basis of .
Let denote the prime field of ; that is, . Suppose that the set is a -linearly independent finite subset of . Let . Then is an additive subgroup of . Let be the truncated polynomial algebra satisfying for all . For every element , define . Then for all . Let denote . Then . Let . Then is an associative superalgebra with -gradation induced by the trivial -gradation of and the natural -gradation of ; that is, , 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, if is a superalgebra (or -graded linear space), let be the derivation superalgebra of (see  or  for the definition) and ; that is, is the set of all -homogeneous elements of . If occurs in some expression, we regard as a -homogeneous element and as the -degree of . Let be a -graded superalgebra. If , then we call a -homogeneous element and the -degree of and set .
Set . Given that , let be the partial derivative on with respect to . For , let be the linear transformation on such that for all and . Then for all since .
Suppose that and . When , we denote the uniquely determined element of satisfying by and denote the number of integers less than in by . When , we set and . Therefore, for any and .
We define for , and . Since the multiplication of is supercommutative, it follows that is a derivation of . Let Then is a finite dimensional Lie superalgebra contained in . A direct computation shows that where , and , .
Let be the linear map such that for every and , where and . It is easy to see that is an even linear map. Let . Then is a finite dimensional Lie superalgebra with a -gradation , where . In this paper, is called the Lie superalgebra of special type.
By the definition of linear map , the following equalities are easy to verify: where , ; , , ; and , and as in (3). The equality (5) shows that is a subalgebra of . Hereafter, and will be simply denoted by and , respectively.
Put and .
Proposition 1. The Lie superalgebra is generated by .
Proof. Suppose that generate the subalgebra of . Since and are subsets of , it follows that .
Next we will consider the reverse inclusion.
It is easy to see that for all distinct elements , of and . Therefore, and .
A direct calculation shows that for all distinct elements , , , of and , . It follows from that .
For distinct elements , , , , of and , , , we have and . Thus .
By the same methods above, we may obtain for ; that is, for .
According to and , we have Hence .
In conclusion, . Therefore, the desired result follows immediately.
3. The Natural Filtration of
Adopting the notion of , the element of Lie superalgebra is called ad-nilpotent if is a nilpotent linear transformation. The set of all ad-nilpotent elements of is denoted by . Let . Then is a descending filtration of , which is called the natural filtration of . We also call a filtration of for short, where if and if . Since is -graded and finite dimensional, we may easily obtain and .
Let denote the set of all matrices over . Notice that . Without loss of generality, we may suppose that is a standard -basis of . If , where , then let , where .
Lemma 2. Suppose that . If is ad-nilpotent, then is a nilpotent matrix.
Proof. Let be the representation of with values in . Then and the matrix of over the basis of is , where . Since is ad-nilpotent, the representation is a nilpotent linear transformation. It implies that is nilpotent. Therefore, is a nilpotent matrix.
Lemma 3. Let , where and . If and , then .
Proof. Suppose that , where and . Since , we may assume that . Let be a -homogeneous element of with -degree . Then . On the other hand, which implies . It is easy to see that and . Thus . Since is an arbitrary -homogeneous element of , we have . Then ; that is, .
Suppose that denotes the matrix whose element is and otherwise is zero. Obviously, where is the Kronecker delta.
If , where , then
Lemma 4. Suppose that , where . If , then .
Proof. Suppose that , where . Let and , , such that , , are distinct. We may assume that . A direct calculation shows that
By equalities (11) and (12), we have Since , we have . So is not a nilpotent matrix. By Lemma 2, it follows that . By Lemma 3, we have . Then . It contradicts . This proves our assertion.
Lemma 5. Let , where . If , then .
Proof. Assume that . Let , , and
(i) Suppose that . Let Then . It is easy to see that . Since , we have . Therefore, Assume that . It follows from that . Then we have which implies that Therefore, where is an matrix and . Since , we have not being a nilpotent matrix. Then is not a nilpotent matrix and . Lemma 3 shows that . It is a contradiction of to ; that is, .
Suppose that and , . Let . By equality (2), we obtain Since , also has the matrix form as , it follows from that is not a nilpotent matrix. Then is not nilpotent. So and . It is a contradiction of .
(ii) Suppose that . Let and . Imitating , we may prove that is also not nilpotent. Then the desired result follows.
If and , then .
Suppose that , are distinct elements of ; then for all .
Suppose that , , are distinct elements of ; then , where , and , are arbitrary elements of .
Proof. (i) A direct verification shows that is a weakly closed subset of nilpotent elements of , where is the general linear Lie superalgebra of . It was shown in [23, Theorem 1 of Chapter II] that each element of is a nilpotent linear transformation of . Then is nilpotent. So is ad-nilpotent.
(ii) To prove , we may assume without loss of generality that . Set to be an arbitrary element of . If , then In the case of , we have
For we obtain . Therefore . This yields . Thus .
(iii) According to and , is a weakly closed subset of nilpotent elements of . So , where , .
Lemma 7. If , , are distinct elements of , then for all .
Lemma 8. .
Proof. By the definition of , we have . Lemmas 4 and 5 show that . Then . Thus .
Next we will prove . Let and , where . Assume that , . Without loss of generality, we may suppose that . Let , where , , are distinct elements of and is an arbitrary element of . By Lemma 7, we have . Let , where and . Since , we have . Lemma 5 implies that . It is a contradiction of . Hence .
Assume that , , and suppose that and are as the definitions in (15). We may suppose that (the proof is similar to the case ) and let be as the definition in (16). In a similar way to the first part of the proof in Lemma 5, we have . Suppose that and . Lemma 7 shows that . Let , where and . Using equality (2), we have If , then in the above equality, where . Thus By equality (12), the matrix has the matrix form as in Lemma 5. Since , is not a nilpotent matrix. It implies that is not nilpotent. Hence . Lemma 3 shows that ; that is, . It contradicts . Thus . Therefore, and .
According to the fact that and are invariant subspaces under the automorphisms of and Lemma 8, is also invariant under the automorphisms of . Since we may easily obtain the following theorem.
Theorem 9. The natural filtration of is invariant under the automorphisms of .
Let for . Then is a -graded space. Suppose that ; then is also a -graded space. Let and . Define It is easy to see that the definition above is reasonable. There exists a linear expansion such that has an operator . A direct verification shows that is a Lie superalgebra with respect to the operator . The Lie superalgebras is called a Lie superalgebra induced by the natural filtration of .
Lemma 10. .
Proof. Let be a linear map such that , where . A direct verification shows that is a homomorphism of Lie superalgebras. Suppose that . If , then there exists such that . Since , we have . Hence . That shows that . Thus, . Therefore, is a monomorphism. It follows from the fact is finite dimensional that is an isomorphism.
The definition of shows that
Suppose that , , , are elements of and , are greater than . In a similar way to , the Lie superalgebra will be simply denoted by . According to the definitions of , , and in , the , , and in are also defined by the same method, respectively.
Proposition 11. Suppose that and is an isomorphism from to ; then for all .
Lemma 12. Suppose that and is an isomorphism from to ; then induces an isomorphism from to such that for all .
Proof. Define a linear map such that
where . Using Proposition 11, the definition of is reasonable and
Thus is a homomorphism from to . Clearly, for all . It follows that is an epimorphism.
Suppose that ; then . So we may suppose that and . Since , let , where . Hence . It follows from that . Thus ; that is, . It follows that . By Proposition 11, we have . Then for . Therefore, and . Consequently, is an isomorphism induced by such that for all .
Theorem 13. if and only if and .
Proof. Because the sufficiency is obvious, it suffices to prove the necessity. Suppose that is the isomorphism given in the proof of Lemma 10. Similarly, there also exists the . According to the equality (28) and Lemma 12, we have for . Let . Then In particular, . It follows from that . By virtue of the definition of , we have Thus . Similarly, . According to and , we have . In conclusion, the proof is completed.
The authors thank Yang Jiang for the helpful comments and suggestions. They also give their special thanks to the referees for many helpful suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 11171055) and the Fundamental Research Funds for the Central Universities (no. 12SSXT139).
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