Quantum Affine Algebras at Roots of Unity

Abstract

We study the restricted form of the qaunatized enveloping algebra of an untwisted affine Lie algebra and prove a triangular decomposition for it. In proving the decomposition we prove several new identities in the quantized algebra, one of these show a connection between the quantized algebra and Young diagrams. These identities are all invisible in the non-quantum case of the problem which was considered by Garland in 1978. We then study the finite-dimensional irreducible representations and prove a factorization theorem for such representations.

54 pages, Amstex. The paper has been substantially revised, reorganized and the results are now proved for arbitrary untwisted affine Lie algebras. The proof uses several new identities in the the quantized enveloping algbera