In mathematical representation theory, a Harish-Chandra homomorphism is a homomorphism from a subalgebra of the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representati ...
of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra. A particularly important special case is the
Harish-Chandra isomorphism
In mathematics, the Harish-Chandra isomorphism, introduced by ,
is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center \mathcal(U(\mathfrak)) of the universal enveloping algebra U(\mathfr ...
identifying the center of the universal enveloping algebra with the invariant polynomials on a Cartan subalgebra.
In the case of the ''K''-invariant elements of the universal enveloping algebra for a maximal compact subgroup ''K'', the Harish-Chandra homomorphism was studied by .
References
*
*{{Citation , last1=Howe , first1=Roger E. , editor1-last=Doran , editor1-first=Robert S. , editor2-last=Varadarajan. , editor2-first=V. S. , title=The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) , url=https://books.google.com/books?id=mk-4pl9IftMC&pg=321 , publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, location=Providence, R.I. , series=Proc. Sympos. Pure Math. , isbn=978-0-8218-1197-9 , mr=1767901 , year=2000 , volume=68 , chapter=Harish-Chandra homomorphisms , pages=321–332
Representation theory of Lie groups