Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras

Abstract

Let $Γ$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(Γ\times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(α,a) \in Γ\times A} {\frak L}_{(α,a)}$ by Lie superalgebra automorphisms preserving the $(Γ\times A)$-gradation. In this paper, we show that the Euler-Poincaré principle yields the generalized denominator identity for ${\frak L}$ and derive a closed form formula for the supertraces $\text{str}(g|{\frak L}_{(α,a)})$ for all $g\in G$,$(α,a) \in Γ\times A$. We discuss the applications of our supertrace formula to various classes of infinite dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac-Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $GL(n) \times GL(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and the irreducible highest weight modules over a generalized Kac-Moody superalgebra ${\frak g}$ corresponding to the Dynkin diagram automorphism $σ$ are the same as the usual characters of Verma modules and irreducible highest weight modules over the orbit Lie superalgebra $\breve{\frak g}={\frak g}(σ)$ determined by $σ$.