Graded Lie superalgebras and super-replicable functions

Abstract

In this paper, we investigate the properties of super-replicable functions and their connections with graded Lie superalgebras. The Euler–Poincaré principle for the homology of graded Lie superalgebras yields a certain product identity called the generalized denominator identity. Applying the formal directional derivatives and the Laplacian, we derive a recursive supertrace formula for the graded Lie superalgebras with gradation-preserving endomorphisms. On the other hand, the super-replicable functions are characterized by certain product identities that have the same form as the generalized denominator identities for some graded Lie superalgebras. We derive many interesting relations among the Fourier coefficients of super-replicable functions and their super-replicates, and compute the supertraces of Monstrous Lie superalgebras associated with super-replicable functions. Finally, we study the properties of the hauptmoduln J_{1,N} of Γ_1(N) for N=5,8,10,12, which are super-replicable functions, and determine the Fourier coefficients of their super-replicates J_{1,N}^{(m)} (m \geq 1).