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 ${\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))}$ of the universal enveloping algebra ${\displaystyle U({\mathfrak {g}})}$ of a reductive Lie algebra ${\displaystyle {\mathfrak {g}}}$ to the elements ${\displaystyle S({\mathfrak {h}})^{W}}$ of the symmetric algebra ${\displaystyle S({\mathfrak {h}})}$ of a Cartan subalgebra ${\displaystyle {\mathfrak {h}}}$ that are invariant under the Weyl group ${\displaystyle W}$.