Free Lie algebras, generalized Witt formula, and the denominator identity

Abstract

Let G be a countable abelian semigroup satisfying a suitable finiteness condition, and let L= directsum of L_a be the free Lie algebra generated by a G-graded vector space V over C. In this paper, from the denominator identity, we derive a dimension formula for the homogeneous subspaces of the free Lie algebra L= directsum of L_a and discuss numerous applications of our dimension formula to various interesting cases. Our dimension formula will be expressed in terms of the Witt partition functions. Various expressions of the Witt partition functions will give rise to a number of interesting combinatorial identities. As a special case, we obtain a recursive relation for the coefficients of the elliptic modular function j.