Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras

摘要

Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abeliangroup satisfying a certain finiteness condition. Suppose that a group $G$ actson a $(\Gamma \times A)$-graded Lie superalgebra ${\frakL}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Liesuperalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In thispaper, we show that the Euler-Poincar\'e principle yields the generalizeddenominator identity for ${\frak L}$ and derive a closed form formula for thesupertraces $\text{str}(g|{\frak L}_{(\alpha,a)})$ for all $g\in G$,$(\alpha,a)\in \Gamma\times A$. We discuss the applications of our supertrace formula tovarious classes of infinite dimensional Lie superalgebras such as free Liesuperalgebras and generalized Kac-Moody superalgebras. In particular, wedetermine the decomposition of free Lie superalgebras into a direct sum ofirreducible $GL(n) \times GL(k)$-modules, and the supertraces of the MonstrousLie superalgebras with group actions. Finally, we prove that the generalizedcharacters of Verma modules and the irreducible highest weight modules over ageneralized Kac-Moody superalgebra ${\frak g}$ corresponding to the Dynkindiagram automorphism $\sigma$ are the same as the usual characters of Vermamodules and irreducible highest weight modules over the orbit Lie superalgebra$\breve{\frak g}={\frak g}(\sigma)$ determined by $\sigma$.