Let g be a finite-dimensional simple Lie algebra and let Lg = g[t,t^{-1}] be the loop algebra of g.  We consider the universal enveloping algebra U(Lg).  This is a noncommutative ring which deforms a polynomial ring in countably many variables, and a priori it is hard to see why U(Lg) should have any finiteness properties -- for example, unlike other enveloping algebras of infinite-dimensional Lie algebras, it is straightforward to see that U(Lg) is neither left or right noetherian.

 

However, it turns out the two-sided structure of U(Lg) is much more constrained.  We show that (two-sided) ideals of U(Lg) are extremely large, in the sense that if J is a nontrivial ideal of U(Lg), then U(Lg)/J is the same size as a polynomial ring in _finitely_ many variables.  As an application, we show that a nontrivial central quotient of the enveloping algebra of an affine Lie algebra is simple.

 

This is joint work with Rekha Biswal.

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