Quantized hyperalgebras of rank 1

2004/03/08 by William Chin, Chin, William, Leonid Krop +1
Mathematics · #17B37 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Rings, Modules, and Algebras #math.QA #msc:17B37

paper · pdf · doi:10.48550/arxiv.math/0403144

arxiv created 2004/03/08 · openalex publication_date 2004/03/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the algebra U_ζ obtained via Lusztig's `integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra \frak g=sl₂ modulo the two-sided ideal generated by K^l-1. We show that U_ζ is a smash product of the quantum deformation of the restricted universal enveloping algebra \bold u_ζ of \frak g and the ordinary universal enveloping algebra U of \frak g, and we compute the primitive (= prime) ideals of \Uz. Next we describe a decomposition of \bold u_ζ into the simple U- submodules, which leads to an explicit formula for the center and the indecomposable direct summands of \Uz. We conclude with a description of the lattice of cofinite ideals of \Uz in terms of a unique set of lattice generators.

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