QUANTIZED HYPERALGEBRA OF RANK 1

Abstract. We study the algebra Uζ obtained via Lusztig’s ‘integral ’ form [Lu 1, 2] of the generic quantum algebra for the Lie algebra g = sl2 modulo the two-sided ideal generated by Kl − 1. We show that Uζ is a smash product of the quantum deformation of the restricted universal enveloping algebra uζ of g and the ordinary universal enveloping algebra U of g, and we compute the primitive ( = prime) ideals of Uζ. Next we describe a decomposition of uζ into the simple U- submodules, which leads to an explicit formula for the center and the indecomposable direct summands of Uζ. We conclude with a description of the lattice of cofinite ideals of Uζ in terms of a unique set of lattice generators. G. Lusztig constructed in [Lu 1,2,3] quantum algebras associated to the defining relations of the finite-dimensional semi-simple Lie algebra g. The method used there is similar to the one employed by Kostant [Ko] in his construction of the hyperalgebra for g. We refer to Lusztig’s algebra as the quantum hyperalgebra o