Harish-Chandra homomorphism
- Harish-Chandra homomorphism
In mathematics, the Harish-Chandra homomorphismis an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps 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".
Let "n" be the rank of "g", which is the dimension of the Cartan subalgebra "h". H. S. M. Coxeter observed that "S"("h")"W" is a polynomial algebra in "n" variables (see Chevalley–Shephard–Todd theorem for a more general statement). Therefore, the center of the universal enveloping algebra of a semisimple Lie algebra is a polynomial algebra. 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 |
|---|
| A"n" | "n"+1 | "n"+1 | 2, 3, 4, ..., "n"+1 |
|---|
| B"n" | 2"n" | 2"n"−1 | 2, 4, 6, ..., 2"n" |
|---|
| C"n" | 2"n" | "n"+1 | 2, 4, 6, ..., 2"n" |
|---|
| D"n" | 2"n"−2 | 2"n"−2 | "n"; 2, 4, 6, ..., 2"n"−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 "G"2 is a polynomial algebra on generators of degrees 2 and 6.
Examples
*If "g" is the Lie algebra "sl"2(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.
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.
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