On Blocks in Restricted Representations of Lie Superalgebras of Cartan Type
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
On Blocks in Restricted Representations of Lie Superalgebras of Cartan Type Fei Fei DUAN1) School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, P. R. China E-mail : [email protected]
Bin SHU School of Mathematical Sciences, East China Normal University, Shanghai 200241, P. R. China E-mail : [email protected]
Yu Feng YAO Department of Mathematics, Shanghai Maritime University, Shanghai 201306, P. R. China E-mail : [email protected] Abstract Let g be a restricted Lie superalgebra of Cartan type W (n), S(n) or H(n) over an algebraically closed field k of prime characteristic p > 3, in the sense of modular version of Kac’s definition in 1977. In this note, we show that the restricted representation category over g has only one block (reckoning parities in). This phenomenon is very different from the case of characteristic zero. Keywords
Lie superalgebra of Cartan type, restricted representation, block
MR(2010) Subject Classification
1
17B10, 17B66, 17B70
Introduction
1.1 As is well known, finite-dimensional simple Lie algebras over an algebraically closed filed of characteristic p > 5 are classified, which are either of classical type, or of Cartan type ([14]). As a counterpart in the super case, Kac’s classification theorem says that all finite-dimensional simple Lie superalgebras over C fall into two types, ones of classical type and ones of Cartan type ([8]). Although we are far away from a complete classification of finite-dimensional simple Lie superalgebras over an algebraically closed field of prime characteristic p, one naturally expects that in such a classification (or to say, modular classification), Lie superalgebras of classical type, and of Cartan type in the same sense as introduced by Kac when he did his classification in [8], will play some crucial roles. Among them, Lie superalgebras of Cartan type consist of ones of W (n), S(n) and H(n), each of them contain infinite series of Lie superalgebras. Each of Received October 22, 2019, accepted March 24, 2020 This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11601116, 11671138 and 11771279), Shanghai Key Laboratory of PMMP (Grant No. 13dz2260400), the Scientific Research Foundation of Hebei Education Department (Grant No. QN2017090) and the Doctoral Foundation of Hebei Normal University (Grant No. L2016B02) 1) Corresponding author
Duan F. F. et al.
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Cartan type Lie superalgebras mentioned above is a certain subalgebra of derivations on some Grassmanian superalgebra. 1.2 In [13], the authors initiated to study representations of Lie superalgebras of Cartan type in prime characteristic, obtaining irreducible character formulas for certain Z-graded representation category of W (n), and their Cartan invariants. Since then, some progress has been made in this direction (see [2, 3, 12, 15, 16] etc.). In particular, in [15] the authors investigated blocks of the restricted repres
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