Restricted Kac modules of Hamiltonian Lie superalgebras of odd type

Abstract

This paper aims to describe the restricted Kac modules of restricted Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic p > 3. In particular, a sufficient and necessary condition for the restricted Kac modules to be irreducible is given in terms of typical weights.

Character Formulas for Simple Modules of Hamiltonian Lie Superalgebras of Odd Type

“…The following lemma is already contained in the proof of [16,Theorem 1]. However, for the reader's convenience, we also give a proof.…”

“…Note that I b i (λ) and L b i (λ) are (u(g), T)-modules, where 0 ≤ i ≤ 2n and λ ∈ F n+1 p (see [16]).…”

“…Note that the latter four series of Lie superalgebras possess more complicated structures and have no analogues in Lie algebra case. In 2014, the authors obtained a sufficient and necessary condition for the restricted Kac modules to be simple for the restricted Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic p > 3 [16].…”

“…Let g be a Hamiltonian Lie superalgebra of odd type over an algebraically closed field of characteristic p > 3. In [16] root reflections are used to construct a series of Borel subalgebras of g and to observe how the highest weights for simple quotients of restricted Kac modules change along with the Borel subalgebras of g (see Lemma 3.1 below). Moreover, a group action on restricted Kac modules of g is also introduced, which is consistent with the module action of Lie superalgebra g itself and then the lengths of simple quotients are determined for the restricted Kac modules of g with atypical weights (see Lemma 3.3).…”

On restricted simple modules of contact Lie superalgebras of odd type

“…Note that the latter four series of Lie superalgebras have no analogues in Lie algebras and each of them possesses a more complicated structure. Liu, Wang and Yuan studied restricted representations of the first two types in the latter four series in [13,17]. This paper is sequel to an early paper [13], in which a sufficient and necessary condition is provided in terms of typical or atypical weights for a Kac module of contact Lie superalgebras of odd type over an algebraically closed field of characteristic > 3 to be simple.…”