On Polynomial Representations of Strange Lie Superalgebras P ( n )
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On Polynomial Representations of Strange Lie Superalgebras P (n ) Cuiling Luo1 Received: 20 June 2018 / Accepted: 20 August 2020 / © Springer Nature B.V. 2020
Abstract In this paper, both canonical and noncanonical polynomial representations of strange classical Lie superalgebra P (n) are investigated. It turns out all these representations are undecomposable, and their composition series are obtained. Keywords Lie superalgebra · Polynomial representation · Minimal modules · Composition series Mathematics Subject Classification (2010) 17B10
1 Introduction Lie superalgebras were introduced by physicists as fundamental tools of studying the supersymmetry in physics. Unlike Lie algebra case, finite-dimensional modules of finitedimensional simple Lie superalgebras may not be completely reducible and the structure of finite-dimensional irreducible modules is much more complicated due to the existence of so-called atypical modules (cf. [19, 20]). In his celebrated work [18] on classification of finite-dimensional simple Lie superalgebras, Kac found two families of exotic classical simple Lie superalgebras, which are called “strange” Lie superalgebras of type P and Q, respectively. These superalgebras do not have analogues in Lie algebras. The strange Lie superalgebras have attracted a number of mathematicians’ attention. Javis and Murray [8] obtained the Casimir invariants, characteristic identities, and tensor operators for the strange Lie superalgebras. Nazarov [17] found Yangians of these superalgebras. In [4], Frappat, Sciarrino and Sorba gave a certain oscillator realization. Gruson [6] computed the Research supported by Natural Science Foundation of Hebei Province of China (A2018209206) and NSFC Grant 11501163 Presented by: Vyjayanthi Chari Cuiling Luo
[email protected] 1
College of Science, North China University of Science and Technology, 21 Bohai Road, Caofeidian Xincheng, Tangshan, Hebei, 063210, China
C. Luo
cohomology with trivial coefficients. For the strange algebras P (n), Frappat, Sciarrino and Sorba [3] studied Dynkin-like diagrams and a certain representation. Medak [14] generalized the Baker-Campbell-Hausdorff formula and used it to examine the so-called BCH-Lie and BCH-invertible subalgebras in P (n). Gorelik [5] obtained the center of the universal enveloping algebra. Serganova [21] determined the center of the quotient algebra of the universal enveloping algebra by its Jacobson radical and used it to study the typical highest weight modules of the algebra. Medak [15] proved that each maximal invertible subalgebra of P (n) is Z-graded. Moon [16] obtained a “Schur-Weyl duality” of P (n). He studied the tensor product of natural representation by the centralizer of P (n) on this tensor product, and decomposed the 2th and 3th tensor. Polynomial representations of finite-dimensional simple Lie algebras are very important from application point of view, where both the representation formulas and bases are clear. It is an example of supsymmetric algebras of the direct sums of natural module an
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