Harish-Chandra homomorphism for generalized Takiff algebras
type: Item
instance of: scholarly article
title: Harish-Chandra homomorphism for generalized Takiff algebras
zbMATH Open document ID: 0847.17006
author: François Geoffriau
published in: Journal of Algebra
publication date: 1996-10-15T00:00:00Z
review text: Let \({\mathfrak g}\) be a semisimple Lie algebra over an algebraically closed field \(k\) of characteristic zero. The generalized Takiff algebra \({\mathfrak g}_\infty\) of \({\mathfrak g}\) is the algebra obtained by extending scalars to the polynomial ring \(k[ T]\) in one variable over \(k\) and then restricting scalars back to \(k\). It is the inverse limit of its quotients \({\mathfrak g}_m:= {\mathfrak g}/ {\mathfrak p}_m\) by the ideals \({\mathfrak p}_m\) generated by \(T^m \otimes {\mathfrak g}\); the first such quotient \({\mathfrak g}_1\) is the (ordinary) Takiff algebra. In previous work the author has shown that the enveloping algebra of \({\mathfrak g}_1\) is a free module over its center (using methods that should extend to all \(m\)) [C. R. Acad. Sci., Paris, Sér. I 319, 11-14 (1994; Zbl 0812.17010)]. In this paper he studies the center itself, showing that it is isomorphic to the ring of invariant functions in the symmetric algebra by symmetrization, while the latter ring is embedded in the ring of Weyl group invariants on the symmetric algebra of a Cartan subalgebra via a map which becomes an isomorphism on passing to the quotient fields.
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 735310
zbMATH Keywords: Harish-Chandra homomorphism
zbMATH Keywords: generalized Takiff algebra
zbMATH Keywords: center
zbMATH Keywords: ring of invariant functions
reviewed by: William M. McGovern
MaRDI profile type: Publication
full work available at URL: jabr.1995.1021
OpenAlex ID: W2066141595
DOI: 10.1006/JABR.1995.1021
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description: scientific article; zbMATH DE number 735310