Harish-Chandra homomorphism
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In mathematics, the Harish-Chandra homomorphism is an isomorphism of commutative rings from the center Z(U(g)) of the universal enveloping algebra U(g) of a semisimple Lie algebra g to the elements S(h)W of the symmetric algebra S(h) of a Cartan subalgebra h that are invariant under the Weyl group W.
In particular, the center of the universal enveloping algebra is a polynomial algebra in n variables, where n is the rank of the semisimple Lie algebra (equal to the dimension of the Cartan subalgebra). The degrees of the generators 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 |
For example, the center of the universal enveloping algebra of G2 is a polynomial algebra on generators of degrees 2 and 6.
[edit] Examples
- If g is the Lie algebra sl2(R), then the center of the universal enveloping algebra is generated by the Casimir invariant of degree 2, and the ring of invariants of the Weyl group is also generated by an element of degree 2.
[edit] References
- Knapp, Vogan, Cohomological induction and unitary representations, ISBN 0-691-03756-6
- Knapp, Anthony, Lie groups beyond an introduction, Second edition, page 300-303.
