Harish-Chandra isomorphism

Isomorphism of commutative rings constructed in the theory of Lie algebras

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 of the universal enveloping algebra of a reductive Lie algebra to the elements of the symmetric algebra of a Cartan subalgebra that are invariant under the Weyl group .

Let be a semisimple Lie algebra, its Cartan subalgebra and be two elements of the weight space (where is the dual of ) and assume that a set of positive roots have been fixed. Let and be highest weight modules with highest weights and respectively.

The modules and 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). So, for and ,

and similarly for , where the functions are homomorphisms from to scalars called central characters.

For any , the characters if and only if and are on the same orbit of the Weyl group of , where is the half sum of the positive roots, sometimes known as the Weyl vector. Another closely related formulation is that the Harish Chandra homomorphism from the center of the universal enveloping algebra to (the elements of the symmetric algebra of the Cartan subalgebra fixed by the Weyl group) is an isomorphism.

The theorem has been used to obtain a simple Lie algebraic proof of Weyl's character formula for finite dimensional irreducible representations. 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
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). A simple consequence is that for Verma modules or generalized Verma modules with highest weight , there exist only finitely many weights for which a non zero homomorphism exists.

For a simple Lie algebra, let be its rank, that is, the dimension of any Cartan subalgebra of H. S. M. Coxeter observed that is isomorphic to a polynomial algebra in variables (see Chevalley–Shephard–Todd theorem for a more general statement). Therefore, the center of the universal enveloping algebra of a simple Lie algebra is isomorphic to a polynomial algebra. The degrees of the generators of the algebra are the degrees of the fundamental invariants given in the following table. Lie algebra Coxeter number h Dual Coxeter number Degrees of fundamental invariants R 0 0 1 An n + 1 n + 1 2, 3, 4, , n + 1 Bn 2n 2n − 1 2, 4, 6, , 2n Cn 2n n + 1 2, 4, 6, , 2n Dn 2n − 2 2n − 2 n; 2, 4, 6, , 2n − 2 E6 12 12 2, 5, 6, 8, 9, 12 E7 18 18 2, 6, 8, 10, 12, 14, 18 E8 30 30 2, 8, 12, 14, 18, 20, 24, 30 F4 12 9 2, 6, 8, 12 G2 6 4 2, 6

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. In particular, if the fundamental invariants have degrees , then the generators of the cohomology ring have degrees 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. The cohomology ring is isomorphic to a polynomial algebra on generators with degrees .

If is the Lie algebra , then the center of the universal enveloping algebra is generated by the Casimir invariant of degree 2, and the Weyl group acts on the Cartan subalgebra, which is isomorphic to , by negation, so the invariant of the Weyl group is the square of the generator of the Cartan subalgebra, which is also of degree 2. For , the Harish Chandra isomorphism says is isomorphic to a polynomial algebra of Weyl invariant polynomials in two variables (since the Cartan subalgebra is two dimensional). For , the Weyl group is which acts on the CSA in the standard representation. Since the Weyl group acts by reflections, they are isometries and so the degree 2 polynomial is Weyl invariant. The contours of the degree 3 Weyl invariant polynomial (for a particular choice of standard representation where one of the reflections is across the x axis) are shown below. These two polynomials generate the polynomial algebra, and are the fundamental invariants for For all the Lie algebras in the classification, there is a fundamental invariant of degree 2, the quadratic Casimir. In the isomorphism, these correspond to a degree 2 polynomial on the CSA. Since the Weyl group acts by reflections on the CSA, they are isometries, so the degree 2 invariant polynomial is where is the dimension of the CSA , also known as the rank of the Lie algebra. For , the Cartan subalgebra is one dimensional, and the Harish Chandra isomorphism says is isomorphic to the algebra of Weyl invariant polynomials in a single variable The Weyl group is acting as reflection, with non trivial element acting on polynomials by The subalgebra of Weyl invariant polynomials in the full polynomial algebra is therefore only the even polynomials, generated by
For , the Weyl group is , acting on two coordinates , and is generated (non minimally) by four reflections, which act on coordinates as Any invariant quartic must be even in both and , and invariance under exchange of coordinates means any invariant quartic can be written Despite this being a two dimensional vector space, this contributes only one new fundamental invariant as lies in the space. In this case, there is no unique choice of quartic invariant as any polynomial with (and not both zero) suffices.