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.