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Harish-Chandra Homomorphism
In mathematical representation theory, a Harish-Chandra homomorphism is a homomorphism from a subalgebra of the universal enveloping algebra of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra. A particularly important special case is the Harish-Chandra isomorphism 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 mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand–N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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(\mathfrak) of a reductive Lie algebra \mathfrak to the elements S(\mathfrak)^W of the symmetric algebra S(\mathfrak) of a Cartan subalgebra \mathfrak that are invariant under the Weyl group W. Introduction and setting Let \mathfrak be a semisimple Lie algebra, \mathfrak its Cartan subalgebra and \lambda, \mu \in \mathfrak^* be two elements of the weight space (where \mathfrak^* is the dual of \mathfrak) and assume that a set of positive roots \Phi_+ have been fixed. Let V_\lambda and V_\mu be highest weight modules with highest weights \lambda and \mu respectively. Central characters The \mathfrak-modules V_\lambda and V_\mu are representations of the universal enveloping algebra U(\mathfrak) and its center acts on the modules by scalar m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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