Harish-Chandra isomorphism

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 {\displaystyle {\mathfrak {g}}} to the elements S ( h ) W {\displaystyle S({\mathfrak {h}})^{W}} of the symmetric algebra S ( h ) {\displaystyle S({\mathfrak {h}})} of a Cartan subalgebra h {\displaystyle {\mathfrak {h}}} that are invariant under the Weyl group W {\displaystyle W} .

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