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 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 ,
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. [1]
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.
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 , defining
The second map is the twist map . On viewed as a subspace of it is defined with 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.
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). 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 . [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 . They are elements of the vacuum affine vertex algebra at critical level , where is the dual Coxeter number for which are annihilated by the positive loop algebra part of , that is,
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:
Notes on the 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 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 ,
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. [1]
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.
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 , defining
The second map is the twist map . On viewed as a subspace of it is defined with 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.
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). 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 . [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 . They are elements of the vacuum affine vertex algebra at critical level , where is the dual Coxeter number for which are annihilated by the positive loop algebra part of , that is,
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:
Notes on the Harish-Chandra isomorphism