✦ LIBER ✦

Invariants of universal enveloping algebras of relatively free lie algebras

✍ Scribed by Vesselin Drensky; Giulia Maria Piacentini Cattaneo

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 674 KB
Volume: 225
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1999.8116

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✦ Synopsis

Let Fro(29) be the relatively free algebra of rank m _> 2 in the nonlocally nilpotent variety 29 of Lie algebras over an infinite field of any characteristic. We study the problem of finite generation of the algebra of invariants of a cyclic linear group G = (g) of finite order invertible in the base field, acting on the universal enveloping algebra U(Fm( 29)). If the matrix g has eigenvalues of different multiplicative orders, then we show that the algebra of invariants U(Fm(29)) 6 is not finitely generated. If all eigenvalues of g are of the same order and 29 is a subvariety of the variety ~c~ of all nilp0tent of class c-by-abelian algebras for some c >__ 1, then the algebra of invariants is finitely generated. On the other hand, for every g which is not a scalar matrix, there exists a variety of Lie algebras 29 such that the algebra U(Fm(29)) c is not finitely generated.

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