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Primitive idempotents of the hyperalgebra for the $r$-th Frobenius kernel of ${\rm SL}(2,k)$

Yutaka Yoshii

Abstract

In this paper we construct primitive idempotents of the hyperalgebra for the $r$-th Frobenius kernel of the algebraic group ${\rm SL}(2,k)$.

Primitive idempotents of the hyperalgebra for the $r$-th Frobenius kernel of ${\rm SL}(2,k)$

Abstract

In this paper we construct primitive idempotents of the hyperalgebra for the -th Frobenius kernel of the algebraic group .

Paper Structure

This paper contains 5 sections, 22 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of $\mathcal{U}$
  4. Primitive idempotents in $\mathcal{U}_1$
  5. Primitive idempotents in $\mathcal{U}_r$

Key Result

Proposition 2.1

For $m,n \in \mathbb{Z}_{\geq 0}$ and $s,t \in \mathbb{Z}$, the following holds in $\mathcal{U}$: (i) ${X^{(m)} Y^{(n)} = \sum_{i=0}^{{\rm min}(m,n)} Y^{(n-i)} {H-m-n+2i \choose i} X^{(m-i)}}$, ${Y^{(m)} X^{(n)} = \sum_{i=0}^{{\rm min}(m,n)} X^{(n-i)} {-H-m-n+2i \choose i} Y^{(m-i)}}$, (ii) ${{H +s

Theorems & Definitions (40)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • proof
  • ...and 30 more