Center of the universal enveloping algebra - Imported Knowlpedia Content

Definition

Let $\mathfrak g$ be a Lie algebra over a field $k$ , and let $U(\mathfrak g)$ be its universal enveloping algebra. The center of the universal enveloping algebra is

Z(U(\mathfrak g))=\{z\in U(\mathfrak g):zu=uz\text{ for every }u\in U(\mathfrak g)\}.

Because $U(\mathfrak g)$ is generated as an algebra by $\mathfrak g$ , it is enough to require $zx=xz$ for every $x\in\mathfrak g$ . This center is a commutative unital subalgebra of $U(\mathfrak g)$ , distinct from the center of the Lie algebra $\mathfrak g$ itself.

Action on modules

Every $z\in Z(U(\mathfrak g))$ acts on a $U(\mathfrak g)$ -module by a module endomorphism. A module has a central character when this action is scalar through an algebra homomorphism $Z(U(\mathfrak g))\to k$ ; in real-reductive representation theory this scalar action is called an infinitesimal character. Scalar action is automatic only under appropriate irreducibility and endomorphism hypotheses, not for an arbitrary module.

Semisimple structure

When $\mathfrak g$ is complex semisimple, the center is much larger than the scalars. The Harish–Chandra isomorphism identifies it with the Weyl-invariant part of a polynomial algebra on a Cartan subalgebra. In particular, $Z(U(\mathfrak g))$ is a polynomial algebra on $\operatorname{rank}\mathfrak g$ algebraically independent generators. The quadratic Casimir element is the most familiar generator in rank one.

Filtration

The standard filtration of $U(\mathfrak g)$ restricts to its center. Under the Poincaré–Birkhoff–Witt associated-graded map, leading terms of central elements become adjoint-invariant polynomial functions. This filtered viewpoint explains why invariant polynomial theory governs the center, although the precise algebra isomorphism requires the Harish–Chandra shift.

References

Jacques Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996. DOI record. Relevant: Chapter 7 on centers and central characters.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. DOI record. Relevant: Chapter V, §5.