Braid groups, infinite Lie algebras of Cartan type and rings of invariants (Q1304872)

The author shows that each element \(\alpha\) 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 \(\alpha=\exp(D)=\text{id}+D+(D^2/2!)+(D^3/3!)+\cdots\), where \(D=D(\alpha)\) is an element of an infinite dimensional Lie algebra \({\mathfrak h}(n)\), and uses the representation \(\alpha=\exp(D)\) to prove results about the ring of invariants for this action of the pure braid group. The Lie algebra \({\mathfrak h}(n)\) is a subalgebra of a graded Lie algebra \({\mathfrak k}(n)\). He also calculates the Poincaré series of the Lie algebra \({\mathfrak k}(n)\) and of certain of its subalgebras, and shows that these Poincaré series are rational.