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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Rings of invariant polynomials for a class of Lie algebras


Author: S. J. Takiff
Journal: Trans. Amer. Math. Soc. 160 (1971), 249-262
MSC: Primary 22.50; Secondary 17.00
DOI: https://doi.org/10.1090/S0002-9947-1971-0281839-9
MathSciNet review: 0281839
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Abstract: Let $ G$ be a group and let $ \pi :G \to GL(V)$ be a finite-dimensional representation of $ G$. Then for $ g \in G,\pi (g)$ induces an automorphism of the symmetric algebra $ S(V)$ of $ V$. We let $ I(G,V,\pi )$ be the subring of $ S(V)$ consisting of elements invariant under this induced action. If $ G$ is a connected complex semisimple Lie group with Lie algebra $ L$ and if Ad is the adjoint representation of $ G$ on $ L$, then Chevalley has shown that $ I(G,L,$Ad$ )$ is generated by a finite set of algebraically independent elements. However, relatively little is known for nonsemisimple Lie groups. In this paper the author exhibits and investigates a class of nonsemisimple Lie groups $ G$ with Lie algebra $ L$ for which $ I(G,L,$Ad$ )$ is also generated by a finite set of algebraically independent elements.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0281839-9
Keywords: Symmetric algebra, ring of invariant polynomials, adjoint representation, polynomial functions, contragredient representation, algebraically independent homogeneous polynomials, Levi decomposition, principal nilpotent element, differential operators, formal power series
Article copyright: © Copyright 1971 American Mathematical Society

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