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 be a connected Lie group with Lie algebra and let *h* be a Lie subalgebra of the complexification *g* of . Let be the annihilator of *h* in and let be the annihilator of in the universal enveloping algebra of *g*. If *h* is the complexification of the Lie algebra of a Lie subgroup of then whenever 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 , the other for holomorphic functions. In the first part the Lie algebra of the closure of is characterized as the annihilator in of , and it is proved that is an ideal in and that where *v* is an abelian subalgebra of . In the second part we consider a complexification *G* of and assume that *h* is the Lie algebra of a closed connected subgroup *H* of *G*. Then we establish that if and only if has many holomorphic functions. This is the case if 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 .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1189550-8

Keywords:
Lie subgroup,
closure,
annihilator,
complexification,
and holomorphic functions,
quasi-affine variety,
reductive and unipotent groups

Article copyright:
© Copyright 1994
American Mathematical Society