In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras.
) and assume that a set of positive roots
are representations of the universal enveloping algebra
and its center acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vector).
to scalars called central characters.
is the half-sum of the positive roots, sometimes known as the Weyl vector.
[1] Another closely related formulation is that the Harish-Chandra homomorphism from the center of the universal enveloping algebra
(the elements of the symmetric algebra of the Cartan subalgebra fixed by the Weyl group) is an isomorphism.
More explicitly, the isomorphism can be constructed as the composition of two maps, one from
is not actually Weyl-invariant, but it can be proven that the twisted character
The theorem has been used to obtain a simple Lie algebraic proof of Weyl's character formula for finite-dimensional irreducible representations.
[2] The proof has been further simplified by Victor Kac, so that only the quadratic Casimir operator is required; there is a corresponding streamlined treatment proof of the character formula in the second edition of Humphreys (1978, pp. 143–144).
Further, it is a necessary condition for the existence of a non-zero homomorphism of some highest weight modules (a homomorphism of such modules preserves central character).
be its rank, that is, the dimension of any Cartan subalgebra
variables (see Chevalley–Shephard–Todd theorem for a more general statement).
The number of the fundamental invariants of a Lie group is equal to its rank.
Fundamental invariants are also related to the cohomology ring of a Lie group.
Due to this, the degrees of the fundamental invariants can be calculated from the Betti numbers of the Lie group and vice versa.
In another direction, fundamental invariants are related to cohomology of the classifying space.
is isomorphic to a polynomial algebra on generators with degrees
[3] The above result holds for reductive, and in particular semisimple Lie algebras.
There is a generalization to affine Lie algebras shown by Feigin and Frenkel showing that an algebra known as the Feigin–Frenkel center is isomorphic to a W-algebra associated to the Langlands dual Lie algebra
[4][5] The Feigin–Frenkel center of an affine Lie algebra
is not exactly the center of the universal enveloping algebra
of the vacuum affine vertex algebra at critical level
which are annihilated by the positive loop algebra
is the affine vertex algebra at the critical level.
The isomorphism in this case is an isomorphism between the Feigin–Frenkel center and the W-algebra constructed associated to the Langlands dual Lie algebra by Drinfeld–Sokolov reduction:
as a polynomial algebra in a finite number of countably infinite families of generators,
is the (negative of) the natural derivative operator on the loop algebra.