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

In this paper we show that each element α of the pure braid group Pn or the pure symmetric automorphism group H(n) of the free group Fn of rank n can be represented as α=exp(D)=id+D+(D2/2!)+(D3/3!)+⋯ , 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[[a1,…,ar]], r=n2−n, which fixes the volume form a1∧⋯∧ar and so h(n) is a subalgebra of Sn2−n, the special Lie algebra of Cartan type. There is a corresponding action of these groups on C[[a1,…,ar]] and C[a1,…,ar]. 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.