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 W.

Introduction and setting

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 \Phi_+ have been fixed. Let V_\lambda and V_\mu be highest weight modules with highest weights \lambda and \mu respectively.

Central characters

The -modules V_\lambda and V_\mu 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 V_\mu, where the functions are homomorphisms from to scalars called central characters.

Statement of Harish-Chandra theorem

For any, the characters if and only if and are on the same orbit of the Weyl group of , where \delta 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.

Explicit isomorphism

More explicitly, the isomorphism can be constructed as the composition of two maps, one from to and another from to itself. The first is a projection. For a choice of positive roots \Phi_+, defining as the corresponding positive nilpotent subalgebra and negative nilpotent subalgebra respectively, due to the Poincaré–Birkhoff–Witt theorem there is a decomposition If is central, then in fact The restriction of the projection to the centre is, and is a homomorphism of algebras. This is related to the central characters by The second map is the twist map. On viewed as a subspace of it is defined with \delta the Weyl vector. Then is the isomorphism. The reason this twist is introduced is that is not actually Weyl-invariant, but it can be proven that the twisted character is.

Applications

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 V_\lambda with highest weight \lambda, there exist only finitely many weights \mu for which a non-zero homomorphism exists.

Fundamental invariants

For a simple Lie algebra, let r 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 r 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. 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.

Examples

Generalization to affine Lie algebras

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. The Feigin–Frenkel center of an affine Lie algebra is not exactly the center of the universal enveloping algebra. They are elements S of the vacuum affine vertex algebra at critical level k = -h^\vee, where h^\vee is the dual Coxeter number for which are annihilated by the positive loop algebra part of, that is, where is the affine vertex algebra at the critical level. Elements of this center are also known as singular vectors or Segal–Sugawara vectors. 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: There is also a description of as a polynomial algebra in a finite number of countably infinite families of generators,, where have degrees and \partial is the (negative of) the natural derivative operator on the loop algebra.

External resources

Notes on the Harish-Chandra isomorphism