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Proceedings of the American Mathematical Society

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

 
 

 

On the left ideal in the universal enveloping algebra of a Lie group generated by a complex Lie subalgebra


Author: Juan Tirao
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
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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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Keywords: Lie subgroup, closure, annihilator, complexification, <!– MATH ${C^\infty }$ –> <IMG WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^\infty }$"> and holomorphic functions, quasi-affine variety, reductive and unipotent groups
Article copyright: © Copyright 1994 American Mathematical Society