Braid groups, infinite Lie algebras of Cartan type and rings of invariants

Abstract

In this paper we show that each element α of the pure braid group P n or the pure symmetric automorphism group H( n) of the free group F n of rank n can be represented as α= exp(D)= id+D+(D 2/2!)+(D 3/3!)+⋯ , where D= D( α) is an element of an infinite-dimensional Lie algebra h(n) . Each such D is a derivation of the power series ring C[[a 1,…,a r]] , r= n 2− n, which fixes the volume form a 1∧⋯∧ a r and so h(n) is a subalgebra of S n 2− n , the special Lie algebra of Cartan type. There is a corresponding action of these groups on C[[a 1,…,a r]] and C[a 1,…,a r] . We use the representation α=exp( D) to prove results about the ring of invariants for this action of the pure braid group. The Lie algebra h(n) is a subalgebra of a graded Lie algebra h(n) ; we also calculate the Poincaré series of the Lie algebra l(n) and of certain of its subalgebras, and show that these Poincaré series are rational.