Graded Lie superalgebras, supertrace formula, and orbit Lie superalgebras. (Q2766379)

Let \(\Gamma\) be a countable abelian semigroup, \(\mathcal A\) a countable abelian group and \(L\) a \((\Gamma\times \mathcal A)\)-graded complex Lie superalgebra. Suppose that \(G\) is a group of graded automorphisms of \(L\). A closed form for the supertrace \(\text{ str}\left(g\mid L_{(\alpha,a)}\right)\) using the Witt partition function is found. In the next section the standard homology theory for \(L\) is developed. It is shown that if \(L\) is a free Lie superalgebra generated by a superspace \(V\), then \(H_1(L)=V\) and \(H_i(L)=0\) if \(i\geqslant 2\). Moreover the Witt formula for the free Lie superalgebra \(L\) and the action of \(\mathfrak{gl}(k,l)\) on \(L\) are considered. A Kostant-type formula for homology of the negative part of a generalized Kac-Moody superalgebra with the group action preserving root space decomposition is also studied.NEWLINENEWLINEIn the final sections the authors consider the Monstrous Lie superalgebra and generalized characters of Verma modules. It is shown that these characters and integrable irreducible highest weight modules over a generalized Kac-Moody superalgebra associated with Dynkin diagram automorphism \(\sigma\) coincide with usual characters of Verma modules and integrable irreducible highest weight modules over an orbit Lie superalgebra determined by \(\sigma\).