# On the Center of the Universal Enveloping of Lie Algebras

Lie algebras are certain non-commutative non-associative structures which are of fundamental importance in mathematics and physics. To study them, one attaches a certain associative algebra to them, known as the universal enveloping algebra. The center of the enveloping algebra is a commutative and associative algebra, and therefore, lends itself to investigation by the methods of algebraic geometry. In addition, it gives significant information about the original Lie algebra and its representations.

In this talk, I will review Harish-Chandra's description of the center in the case that the Lie algebra is semisimple. I will then discuss an analogous theorem of Feigin and Frenkel on the structure of the center for affine Kac-Moody algebras. The geometric structures emerging from the latter theorem have surprising relationship to other areas of mathematics, in particular, to connections on vector bundles, and the Langlands program.