Graded modules over classical simple Lie algebras with a grading

Abstract

Abstract. Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D 4 , where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.

Gradings on the simple real Lie algebras of types G2 and D4

“…We will say that a grading on (R, ϕ) is of Type I if the image of the corresponding homomorphism G D → Aut R (R, ϕ) is contained in the connected component of Aut R (R, ϕ), and of Type II otherwise. The distinction can be made by looking at the image of the character group G := Hom(G, C × ) (i.e., the group of C-points of G D ) under the homomorphism G → PGO 8 (C) associated to the grading on the complexification of (R, ϕ) -see [EK15a,Lemma 33].…”

Graded simple Lie algebras and graded simple representations

“…Grading on simple Lie algebras are, to some extent, well-studied, see [6,7], and we will not study this problem in the present paper. Instead, now we assume that g is not simple.…”

“…We note that a special case of Theorem 31 was obtained in [6] with a totally different approach. We remark that neither P nor V in Theorem 31 are uniquely determined.…”

“…Some results in this direction can be found in [6]. We note that [6,Theorem 8] determined the necessary and sufficient conditions for two finite dimensional M(Q, P, V ) to be graded isomorphic if g is a finite dimensional simple Lie algebra. We complete the paper with the following question: Problem 36.…”

Gradings on Modules Over Lie Algebras of E Types

“…Finite-dimensional G-graded L-modules for semisimple Lie algebras have been studied in [EK15a] and [EK15b,Appendix A]. Every such module is a direct sum of graded simple modules, i.e., G-graded modules that do not contain any proper nonzero graded submodule.…”