Kang, Seok-Jin; Kim, Myung-Hwan Free Lie algebras, generalized Witt formula, and the denominator identity. (English) Zbl 0862.17001 J. Algebra 183, No. 2, 560-594 (1996). 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 \(j(q)\). Reviewer: G.Melançon (Bordeaux) Cited in 16 Documents MSC: 17B01 Identities, free Lie (super)algebras 11F22 Relationship to Lie algebras and finite simple groups Keywords:dimension formula; homogeneous subspace; free Lie algebra; denominator identity; Witt formula; elliptic modular function PDFBibTeX XMLCite \textit{S.-J. Kang} and \textit{M.-H. Kim}, J. Algebra 183, No. 2, 560--594 (1996; Zbl 0862.17001) Full Text: DOI Link