Advances in Matrices, Finite and Infinite, with Applications 2014View this Special Issue
Research Article | Open Access
Qiang Mu, "Natural Filtrations of Infinite-Dimensional Modular Contact Superalgebras", Journal of Applied Mathematics, vol. 2014, Article ID 601847, 9 pages, 2014. https://doi.org/10.1155/2014/601847
Natural Filtrations of Infinite-Dimensional Modular Contact Superalgebras
The natural filtration of the infinite-dimensional contact superalgebra over an algebraic closed field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements and the subalgebras generated by certain ad-nilpotent elements. Moreover, we obtain an intrinsic characterization of contact superalgebras and a property of automorphisms of these Lie superalgebras.
Filtration structures play an important role in the classification of modular Lie algebras (see [1, 2]) and nonmodular Lie superalgebras (see [3, 4]), respectively. We know that the Lie algebras and Lie superalgebras of Cartan type possess a natural filtration structure. The natural filtrations of finite-dimensional modular Lie algebras of Cartan type were proved to be invariant in [5, 6]. In the infinite-dimensional case, the same conclusion was proved in , by determining -nilpotent elements. In the case of Lie superalgebras of Cartan type, the invariance of the natural filtrations of some Lie superalgebras was proved in [8, 9]. Similar results for Lie superalgebras of generalized Cartan type were obtained in [10–12], respectively.
In this paper, we consider the infinite-dimensional modular contact superalgebra , which is analogous to the one in the nonmodular situation (see ). But since the principal -gradations of Lie superalgebras of Cartan type are different (see ), most results and proofs for other Lie superalgebras cannot be applied to contact superalgebras. Therefore the corresponding results and proofs for contact superalgebras have to be established separately. By determining the -nilpotent elements and subalgebras generated by certain -nilpotent elements, we prove the main result of this paper.
Theorem 1. The natural filtration of the infinite-dimensional contact Lie superalgebra is invariant under automorphisms.
Thereby, one obtains the following theorems.
Theorem 2. Suppose that , , , are positive integers. Then if and only if .
Theorem 3. Let , be automorphisms of . Then if and only if .
The paper is organized as follows. In Section 2, we recall the necessary definitions concerning the modular contact superalgebra . In Section 3, we study the ad-nilpotent elements of . In Section 4, we complete the proofs of Theorems 1–3.
Throughout this paper, denotes an algebraic closed field of characteristic and the ring of integers modulo 2. Let and denote the sets of positive integers and nonnegative integers, respectively. Given , and , we put . Let denote the divided power algebra over with basis . For , , we abbreviate to . Let be the exterior superalgebra over in variables , where . Denote by the tensor product . The trivial -gradation of and the natural -gradation of induce a -gradation of such that is an associative superalgebra. For and , we abbreviate to . For and , the following formulas in hold: where
Put , , and . Let where . Given , set , , and . Then is an -basis of the infinite-dimensional superalgebra .
Let be the linear transformations of such that Then are superderivations of the superalgebra . Let Then is an infinite-dimensional Lie superalgebra contained in (see ). If appears in some expression in this paper, we always regard as a -homogeneous element and as the -degree of . Then , where The following formula holds in (see ): where and .
Hereafter let be a positive integer and let . Put and . For , define Let be the linear mapping such that where Then where (see ). It follows directly from (11) and the injectivity of that defines a Lie multiplication on . This Lie superalgebra, denoted by , is called the infinite-dimensional contact superalgebra. In the sequel, we simply write for .
The following formula holds in (see ): Then is a -graded Lie superalgebra, where Let for . Then is referred to as the natural filtration of .
Lemma 4. is transitively graded.
Proof. Assume the contrary, then there exists such that for all , where . Suppose that the largest exponent of among the nonzero summands in the expression of is equal to , and write where . Hence, for , where each exponent of of all nonzero summands in the expression of is less than . Then . Note that all nonzero summands of are -linear independent. It follows that each is equal to . Hence the exponents of in each are equal to . Similarly, for , we can prove that does not appear in each . Consequently, we see that all are of the form . If , then , contradicting . Hence , and we can write where . Note that, for , where each exponent of of the nonzero summands in the expression of is less than . Therefore, Since appears in and does not appear in , we conclude that , a contradiction.
3. ad-Nilpotent Elements
Recall that is called -nilpotent if there exists such that . For a subset of , let denote the set of -nilpotent elements in , and let denote the subalgebra of generated by .
Lemma 5. Suppose that for . The following statements hold.(1)If , then .(2)If , then .(3)If , then .(4)If , then .(5)If , then .
Proof. See [15, Lemma 5.1].
Suppose that . As is -nilpotent, is -nilpotent by . Note that for all . This shows that is not -nilpotent, a contradiction.
By , we see that is -nilpotent. Suppose that , where . Then there exists some . A direct calculation shows that for all . It follows that is not -nilpotent.
is an immediate consequence of , , and . follows from , , and the proof of .
Let and be the -adic expression of , where . Then, is said to be the -adic sequence of , where for all . For , define the -adic matrix of to be As is an matrix with finitely many nonzero elements, is well defined. Let for and . We abbreviate to .
Suppose that is a nonzero element of , where . Define
Given that and , we define
Lemma 6. Let , , and . Then,(1) if and only if ;(2)If , then ;(3)If and , then .
Proof. See [7, Lemma 2.5].
First consider the case . Then, It follows that and , and thus holds.
Next consider the case . We may assume that where and . Hence, If , then If , noting that , then If , then
The assumption implies that . Then it is trivially verified that holds.
Lemma 7. Suppose that , , , and . The following statements hold.(1)If , then .(2)If , then .(3)If , then .
Lemma 8. Suppose that , , and . Let be a nonzero summand of . Then .
Proof. A direct calculation shows that fulfills the conditions of Lemma 7.
Given that , let Clearly, the inequality holds for all .
Lemma 9. .
Proof. Suppose that is an arbitrary element of , where and . Let such that ht. Let be a standard basis element of . By using Lemma 8 repeatedly, we see that .
Lemma 10. For , the following statements hold.(1).(2)If and , then .(3)If and , then .(4).
Proof. Let be a standard basis element of . A direct calculation shows that
Since and , it follows from the binomial theorem that .
Since and , we have .
It follows from that .
Lemma 11. Suppose that . The following statements hold.
Let be distinct elements of , and let be such that . Then .
holds for all distinct .
Proof. A direct calculation shows that . For simplicity, we denote by , respectively. Clearly,
By , we obtain
It follows that
thus proving that .
Let such that . Since , we have . Let . Then yields Hence .
Lemma 12. Let and . Then .
Proof. A direct calculation shows that Since we have
Lemma 13. .
If , then .
If , then .
Proof. Let be an arbitrary element of . Suppose that . Since and , it follows that is not -nilpotent, a contradiction. Hence , and therefore . Noting that is a subalgebra of , we obtain .
Conversely, Lemma 10 shows that , and Lemma 12 implies that . Moreover, since for all , we have . Therefore .
It is clear that , which combined with Lemma 9 yields .
On the other hand, suppose that is an arbitrary element of , where and . By Lemma 5, we have , and hence . Since is a subalgebra of , it follows that .
By , we see that , . The reverse inclusion follows from Lemmas 10 and 11.
Clearly the statement holds when . Now we consider the case . By , we can suppose that is an arbitrary element of , where . If , a direct calculation shows that for all , contradicting that is -nilpotent. Hence and , proving . Since is a subalgebra of , it follows that . The reverse inclusion follows from Lemma 10.
is completely analogous to the proof of .
Let be the corresponding representation with respect to -module ; that is, , . It is easy to see that is faithful. For , we also denote by the matrix of relative to the fixed ordered -basis as follows: Denote by the general linear Lie superalgebra of matrices over . Let denote the matrix whose -entry is and elsewhere, and , where is unit matrix. Let be the Lie algebra consisting of all matrices over satisfying , where is the transpose of . Set ; here
Lemma 14. .
If , then is a nilpotent matrix.
If is an antisymmetric matrix over , we write to denote . As , it is clear that if is a nilpotent matrix then .
Lemma 15. Suppose that . Let be an antisymmetric matrix over of order . If satisfies the following properties:(1);(2) holds for every antisymmetric matrix .
Proof. Let . Three cases arise as follows.
Case 1 . By the property of , we see that