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

✍ Scribed by Stephen P. Humphries

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 260 KB
• Volume: 95
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00007-8

✦ Synopsis

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

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 l(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.