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Quantized hyperalgebras of rank 1

William Chin·Leonid Krop
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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 $\frak {g=sl}_2$ 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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William Chin, Leonid Krop. 2004-03-08. Quantized hyperalgebras of rank 1. https://arxiv.org/abs/math/0403144

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