Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras

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 × A)-graded Lie superalgebra L = ⊕ (α,a)∈Γ×A L(α,a) by Lie superalgebra automorphisms preserving the (Γ × A)-gradation. In this paper, we show that the Euler-Poincaré principle yields the generalized denominator identity for L and derive a closed form formula for the supertraces str(g|L(α,a)) for all g ∈ G, (α, a) ∈ Γ×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)×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 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 ğ = g(σ) determined by σ. ∗Supported in part by Basic Science Research Institute Program, Ministry of Education of Korea, BSRI-98-1414, and GARC-KOSEF at Seoul National University.