Simple Lie algebras in characteristic 2 recovered from superalgebras and on the notion of a simple finite group

Kochetkov, Yuri; Leites, Dimitry

doi:10.1090/conm/131.2/1175822

Cited by 25 publications

(22 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A Lie superalgebra g = g 0 ⊕ g 1 over F is referred to as restricted one, if g 0 is restricted Lie algebra and the g 0 -module g 1 is restricted (see [13,14]). In the following zd(x) denotes the Z-degree of a Z-homogeneous element x.…”

Section: Wen-de Liu and Yong-zheng Zhangmentioning

confidence: 99%

A family of transitive modular Lie superalgebras with depth one

Liu

Zhang

2007

SCI CHINA SER A

View full text Add to dashboard Cite

The embedding theorem is established for Z-graded transitive modular Lie superalgebras g = ⊕ −1 i r gi satisfying the conditions:(i) g0 p(g−1) and g0-module g−1 is isomorphic to the natural p(g−1)-module;(ii) dim g1 = 2 3 n(2n 2 + 1), where n = 1 2 dim g−1. In particular, it is proved that the finite-dimensional simple modular Lie superalgebras satisfying the conditions above are isomorphic to the odd Hamiltonian superalgebras. The restricted Lie superalgebras are also considered. Keywords: flag, divided power algebra, modular Lie superalgebra, embedding theorem MSC(2000): 17B50, 17B05 IntroductionWe know that the Z-graded transitive Lie (super) algebras with depth one play an important role in the classification problem (see [1][2][3][4] for examples). In this paper we are interested in describing a family of the transitive modular Lie superalgebras g = ⊕ −1 i r g i satisfying the conditions in the abstract. More precisely, we shall establish the embedding mappings from these transitive Lie superalgebras into the so-called odd Hamiltonian superalgebras over the fields of prime characteristic, which are analogous to the Leites superalgebras [5,6] or the odd Hamiltonian superalgebras [2] in characteristic zero; they are also called by physicists the Batalin-Vilkoviski algebras. In a modular case, following [2] we call these Lie superalgebras the odd Hamiltonian modular superalgebrs (see sec. 2 for a definition), denoted by H. We should mention that the finite-dimensional simple odd Hamiltonian modular superalgebras have no analogs in the Lie algebras or in the finite-dimensional simple Lie superalgebras in characteristic 0 (cf. [1,2,7]). However, the present paper is motivated by the results and the methods in the modular Lie algebra case and the authors benefit much from reading [3,8,9].As well known, the theory of the Lie superalgebras in characteristic zero has seen a remarkable evolution (see [1,2,10], for examples). But the results in the modular Lie superalgebra case seem not so plentiful. For example, the classification problem remains open for the finite-dimensional

show abstract

Section: Wen-de Liu and Yong-zheng Zhangmentioning

confidence: 99%

A family of transitive modular Lie superalgebras with depth one

Liu

Zhang

2007

SCI CHINA SER A

View full text Add to dashboard Cite

show abstract

“…The research on modular Lie superalgebras just began in recent years (see [11], [15]). The complete classification of the finite-dimensional simple modular Lie superalgebras remains an open problem [12].…”

Section: Introductionmentioning

confidence: 99%

Some properties of the family Γ of modular Lie superalgebras

Chen

2013

Czech Math J

View full text Add to dashboard Cite

“…Kac of finite-dimensional simple Lie superalgebras and infinite-dimensional simple linearly compact Lie superalgebras over algebraically closed fields of characteristic zero have been completed (see [2], [3]). For modular Lie superalgebras, as far as we know, [4] and [10] may be the earliest papers.…”

Section: Introductionmentioning

confidence: 99%

“…For example, the classification problem is still open for finite-dimensional simple Lie superalgebras. As far as we know, the references [4,10] should be the earliest papers on modular Lie superalgebras.…”

Section: Introductionmentioning

confidence: 99%

Derivations of the even part of the odd hamiltonian superalgebra in modular case

Liu

Hua

2009

Acta. Math. Sin.-English Ser.

View full text Add to dashboard Cite

In this paper we mainly study the derivations for even part of the finite-dimensional odd Hamiltonian superalgebra HO over a field of prime characteristic. We first give the generator set of HO 0 . Then we determine the homogeneous derivations from HO 0 into W 0 , the even part of the generalized Witt superalgebra W . Finally, we determine the derivation algebra and outer derivation algebra of HO 0 .Derivations for the even part of the odd Hamiltonian superalgebra 2 was determined for the finite-dimensional odd Hamiltonian superalgebra HO. Note that the derivations of the even parts have been studied for the Lie superalgebras of Cartan type W, S, and H (see [5,6]). However, in contrast to the setting of H, we determine completely the derivation space from the even part of HO into the even part of W, and the derivation algebra of the even part of HO, also. This paper is arranged as follows. In Section 1 we introduce the necessary notations, definitions and known results. In Section 2 we first give the generating set of the even part of the odd Hamiltonian superalgebra. Then we determine the nonnegative Z-degree derivations from the even part of the odd Hamiltonian superalgebra into the even part of the generalized Witt superalgebra. In Section 3 we determine the negative Z-degree derivations from the even part of the odd Hamiltonian superalgebra into the even part of the generalized Witt superalgebra. In Section 4 we determine completely the derivation algebra and outer derivation algebra of the even part of the odd Hamiltonian superalgebra.

show abstract

Simple Lie algebras in characteristic 2 recovered from superalgebras and on the notion of a simple finite group

Cited by 25 publications

References 0 publications

A family of transitive modular Lie superalgebras with depth one

A family of transitive modular Lie superalgebras with depth one

Some properties of the family Γ of modular Lie superalgebras

Derivations of the even part of the odd hamiltonian superalgebra in modular case

Contact Info

Product

Resources

About