Dimension formula for graded Lie algebras and its applications

Abstract

Abstract. In this paper, we investigate the structure of infinite dimensional Lie algebras L = α∈Γ Lα graded by a countable abelian semigroup Γ satisfying a certain finiteness condition. The Euler-Poincaré principle yields the denominator identities for the Γ-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces Lα (α ∈ Γ). Our dimension formula enables us to study the structure of the Γ-graded Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also discuss the relation of graded Lie algebras and the product identities for formal power series.

An analogue to the Witt identity

“…In this section we relate our previous results with Lie algebras and solve Sherman's problem. The solution is provided by the following proposition by S. -J. Kang and M. -H. Kim in [10].…”

“…In [10] S-J. Kang and M-H. Kim derived dimension formulas for the homogeneous spaces of general free graded Lie algebras.…”

On the structure of indefinite quasi-hyperbolic Kac-Moody algebras QHA_2^{(1)}

“…We first give a realization for (1) 2 QHA as a graded Lie algebra of Kac-Moody type and then using the homological techniques developed by Benkart et al, [1] and Kang [6,7,8,9,10] we compute the homology modules of . QHA (1)…”

“…Determination of the structure and multiplicities of roots of higher levels for Kac-Moody algebras is still an open problem. Feingold and Frenkel [2] computed level 2 root multiplicities for the hyperbolic Kac-Moody algebra (1) 1EHA and Kac et al [5] computed root multiplicities for 10 E , Kang [6,7,8,9,10] has computed root multiplicities for roots upto level 5 for HA . As far as the hyperbolic Kac-Moody algebras are concerned, Sthanumoorthy and Uma Maheswari [12] have computed the multiplicities of roots for a particular class of extended hyperbolic Kac -Moody algebra (1) 1…”

The formal series Witt transform

“…In the Lie algebra context the cyclotomic identity is interpreted as a denominator identity related to the free Lie algebra, see e.g. [13,14]. As the referee pointed out the symmetric polynomials in (3) have been studied in the theory of symmetric functions and have applications to counting permutations with certain properties, for example unimodal permutations, see e.g.…”