# Explicit generators of $Z(U(\mathfrak{g}))$

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field. By Harish-Chandra, the center of its universal enveloping algebra $Z(U(\mathfrak{g}))$ is a polynomial ring and the degree of any set of homogeneous generators is uniquely determined.

Where can I find a table describing such generators, e.g. for Lie algebra's of classical type?

• This is a duplicate question, I believe: mathoverflow.net/questions/73836/how-to-find-casimir-operators ; although that question is more general, since the Lie algebra need not be semisimple. – José Figueroa-O'Farrill Feb 17 '15 at 20:20
• There are other questions about such Casimir operators for semisimple Lie algebras, which you should search for. While Harish-Chandra showed how to identify the center of the universal enveloping algebra with an algebra of Weyl group invariants, Chevalley proved (for all finite reflection groups including Weyl groups) that the latter algebra is a polynomial algebra on rank($\mathfrak{g}$) homogeneous generators of uniquely determined degrees. While reflection group invariants of the correct degrees have been worked out case-by-case, it's not easy to "lift" them to the center. – Jim Humphreys Feb 17 '15 at 21:09

A further reference is the thesis of O. Motsak, on "Computation of the central elements and centralizers of sets of elements in non-commutative polynomial algebras". There the center of the universal enveloping algebra is computed explicitly for several examples of classical Lie algebras. The computations have been done in SINGULAR.

• Computations of this sort are necessarily limited but could be valuable if they suggest general patterns. In any case, there apparently doesn't exist any "table" for the classical (or even the exceptional) types as requested by Jack. – Jim Humphreys Feb 18 '15 at 22:40