Graded Lie superalgebras, supertrace formula,and orbit Lie superalgebras

Abstract

Let 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 x A-graded Lie superalgebra L = (, a) x A L(, a) by Lie superalgebra automorphisms preserving the x 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(, a) for all g G, where (, a) x 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) x 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 irreducible highest-weight modules over a generalized Kac–Moody superalgebra g corresponding to the Dynkin diagram automorphism are the same as the usual characters of Verma modules and irreducible highest-weight modules over the orbit Lie superalgebra = g() determined by . 1991 Mathematics Subject Classification: 17A70, 17B01, 17B65, 17B70, 11F22.