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), 12571266
MSC:
Primary 22E15; Secondary 17B35, 22E60
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 quasiaffine 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:
http://dx.doi.org/10.1090/S00029939199411895508
PII:
S 00029939(1994)11895508
Keywords:
Lie subgroup,
closure,
annihilator,
complexification,
and holomorphic functions,
quasiaffine variety,
reductive and unipotent groups
Article copyright:
© Copyright 1994
American Mathematical Society
