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Harish-Chandra isomorphism - Wikipedia

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In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center 𝑍 ( 𝑈 ( 𝑔 ) ) of the universal enveloping algebra 𝑈 ( 𝑔 ) of a reductive Lie algebra 𝑔 to the elements 𝑆 ( ℎ ) 𝑊 of the symmetric algebra 𝑆 ( ℎ ) of a Cartan subalgebra ℎ that are invariant under the Weyl group 𝑊 . Let 𝑔 be a semisimple Lie algebra, ℎ its Cartan subalgebra and 𝜆 , 𝜇 ∈ ℎ ∗ be two elements of the weight space (where ℎ ∗ is the dual of ℎ ) and assume that a set of positive roots Φ + have been fixed. Let 𝑉 𝜆 and 𝑉 𝜇 be highest weight modules with highest weights 𝜆 and 𝜇 respectively. The 𝑔 -modules 𝑉 𝜆 and 𝑉 𝜇 are representations of the universal enveloping algebra 𝑈 ( 𝑔 ) and its center acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vect

Harish-Chandra isomorphism - Wikipedia Jump to content From Wikipedia, the free encyclopedia Isomorphism of commutative rings constructed in the theory of Lie algebras Not to be confused with Harish-Chandra homomorphism . In mathematics , the Harish-Chandra isomorphism , introduced by Harish-Chandra ( 1951 ) , is an isomorphism of commutative rings constructed in the theory of Lie algebras . The isomorphism maps the center Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} of the universal enveloping algebra U ( g ) {\displaystyle U({\mathfrak {g}})} of a reductive Lie algebra g

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