Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras

TL;DR

This paper develops a supertrace formula for graded Lie superalgebras with group actions, deriving identities and applications to infinite-dimensional superalgebras, including free and Kac-Moody types, and relates characters of modules over orbit superalgebras.

Contribution

It introduces a supertrace formula for graded Lie superalgebras with group actions and applies it to various classes, including free and Kac-Moody superalgebras, revealing new structural insights.

Findings

Derived a supertrace formula for graded Lie superalgebras with automorphisms.

Applied the formula to decompose free Lie superalgebras into irreducible modules.

Established equivalence of characters for modules over orbit Lie superalgebras.

Abstract

Let $\Gamma$ 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 $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In this paper, we show that the Euler-Poincar\'e principle yields the generalized denominator identity for ${\frak L}$ and derive a closed form formula for the supertraces $\text{str}(g|{\frak L}_{(\alpha,a)})$ for all $g\in G$,$(\alpha,a) \in \Gamma\times A$. 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 $\sigma$ are the same as the usual characters of Verma modules and irreducible highest weight modules over the orbit Lie superalgebra $\breve{\frak g}={\frak g}(\sigma)$ determined by $\sigma$.