Free Lie algebras, generalized Witt formula, and the denominator identity
The main purpose of the paper is to derive a closed form dimension formula for the homogeneous subspace of a free Lie algebra generated by a vector space \(V\) over \(\mathbb{C}\). The authors also discuss a number of applications of their dimension formula for various interesting cases. The essential ingredient needed is the denominator identity. For the free Lie algebra, it specializes to the well-known Witt formula: \(\prod^\infty_{n=1} (1+q^n)^{L_n} = 1-rq\), where \(L_n\) is the dimension of the homogeneous component of degree \(n\) and \(r\) is the dimension of \(V\). The authors consider the more general framework of a vector space graded over a countable abelian semigroup \(\Gamma\) satisfying suitable finiteness conditions, and suppose moreover that each graded component is of finite dimension. They obtain a generalization of the dimension formula involving Witt partition functions associated with the partitions of the elements in \(\Gamma\). Two special cases are given attention. They first look at the case where \(\Gamma = \mathbb{Z}_{>0}\) is the set of (positive) integers and extend this to a two-dimensional generalization with \(\Gamma = \mathbb{Z}_{>0} \times \mathbb{Z}_{>0}\). They conclude by giving an identity for the coefficients of the elliptic modular function \(j(q)\).
scientific article; zbMATH DE number 923698