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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the left ideal in the universal enveloping algebra of a Lie group generated by a complex Lie subalgebra
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by Juan Tirao PDF
Proc. Amer. Math. Soc. 121 (1994), 1257-1266 Request permission

Abstract:

Let ${G_0}$ be a connected Lie group with Lie algebra ${g_0}$ and let h be a Lie subalgebra of the complexification g of ${g_0}$ . Let ${C^\infty }{({G_0})^h}$ be the annihilator of h in ${C^\infty }({G_0})$ and let $\mathcal {A} = \mathcal {A}({C^\infty }{({G_0})^h})$ be the annihilator of ${C^\infty }{({G_0})^h}$ in the universal enveloping algebra $\mathcal {U}(g)$ of g. If h is the complexification of the Lie algebra ${h_0}$ of a Lie subgroup ${H_0}$ of ${G_0}$ then $\mathcal {A} = \mathcal {U}(g)h$ whenever ${H_0}$ is closed, is a known result, and the point of this paper is to prove the converse assertion. The paper has two distinct parts, one for ${C^\infty }$, the other for holomorphic functions. In the first part the Lie algebra ${\bar h_0}$ of the closure of ${H_0}$ is characterized as the annihilator in ${g_0}$ of ${C^\infty }{({G_0})^h}$, and it is proved that ${h_0}$ is an ideal in ${\bar h_0}$ and that ${\bar h_0} = {h_0} \oplus v$ where v is an abelian subalgebra of ${\bar h_0}$. In the second part we consider a complexification G of ${G_0}$ and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that $\mathcal {A}(\mathcal {O}{(G)^h}) = \mathcal {U}(g)h$ if and only if $G/H$ has many holomorphic functions. This is the case if $G/H$ is a quasi-affine variety. From this we get that if H is a unipotent subgroup of G or if G and H are reductive groups then $\mathcal {A}({C^\infty }{({G_0})^h}) = \mathcal {U}(g)h$.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 1257-1266
  • MSC: Primary 22E15; Secondary 17B35, 22E60
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1189550-8
  • MathSciNet review: 1189550