Z-Graded module of Lie superalgebra H(n) of Cartan type

Let F be an algebraically closed field and charF=0. In this note, using the method ofmixed product in ref. [1], the irreducible positive （negative） Z-graded module of Liesuperalgebra H（n） with base space V （top space V） is determined, where the highest weightof H（n）0-module V is not kλ1 for any nonnegative integer k.

Minuscule representations and Panyushev conjectures Dedicated to the Memory of Professor Bertram Kostant

Recently, Panyushev(2015) raised five conjectures concerning the structure of certain root posets arising from Z-gradings of simple Lie algebras. This paper aims to provide proofs for four of them. Our study also links these posets with Kostant-Macdonald identity, minuscule representations, Stembridge's "t =-1 phenomenon", and the cyclic sieving phenomenon due to Reiner et al.(2004).

Superderivation Algebras of Modular Lie Superalgebras of ■-Type

The authors consider a family of finite-dimensional Lie superalgebras of C-type over an algebraically closed field of characteristic p 〉 3. It is proved that the Lie superalgebras of C-type are simple and the spanning sets are determined. Then the spanning sets are employed to characterize the superderivation algebras of these Lie superalgebras. Finally, the associative forms are discussed and a comparison is made between these Lie superalgebras and other simple Lie superalgebras of Cartan type.

Regularity of vertex operator superalgebras

We prove that the regularity of a vertex operator superalgebra can be reduced to the semisimplicity of the category of its weak modules.Moreover,the rationality can be replaced by requiring that each 1/2Z+-graded module is a direct sum of irreducible weak modules.

Central extensions of generalized orthosymplectic Lie superalgebras

We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra ospm 12n（R,-） that is coordinatized by an arbitrary unital associative superalgebra （R,-） with superinvolution. As a result, an identification between the second homology group of the Lie superalgebra ospm｜2n （R,-） and the first skew-dihedral homology group of the associative superalgebra （R,-） with superin-volution is created for positive integers m and n with （m, n）≠ （1, 1） and （m, n）≠（2, 1）. The second homology groups of the Lie superalgebras ospm1｜2（R,-） and ospm｜2n （R,-） are also characterized explicitly.