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Certain linear isomorphisms for hyperalgebras relative to a Chevalley group

Yutaka Yoshii

Abstract

Let $G$ be a simply connected and simple algebraic group defined and split over a finite prime field $\mathbb{F}_p$ of $p$ elements. In this paper, using an $\mathbb{F}_p$-linear map splitting Frobenius endomorphism on a hyperalgebra relative to $G$, we obtain some $\mathbb{F}_p$-linear isomorphisms induced by multiplication in the hyperalgebra.

Certain linear isomorphisms for hyperalgebras relative to a Chevalley group

Abstract

Let be a simply connected and simple algebraic group defined and split over a finite prime field of elements. In this paper, using an -linear map splitting Frobenius endomorphism on a hyperalgebra relative to , we obtain some -linear isomorphisms induced by multiplication in the hyperalgebra.
Paper Structure (5 sections, 20 theorems, 137 equations)

This paper contains 5 sections, 20 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Commutation formulas in $\mathcal{U}_{\mathbb{Z}}$ and $\mathcal{U}$
  4. Some linear isomorphisms for $\mathcal{U}^+$
  5. Some linear isomorphisms for $\mathcal{U}$

Key Result

Proposition 2.1

Let $\alpha, \beta \in \Phi$, $c \in \mathbb{Z}$, and $m,n \in \mathbb{Z}_{\geq 0}$. In $\mathcal{U}_{\mathbb{Z}}$, the following equalities hold. (i) ${e_{\alpha}^{(m)} e_{\alpha}^{(n)} = {m+n \choose n} e_{\alpha}^{(m+n)}}$. (ii) ${e_{\alpha}^{(m)} e_{-\alpha}^{(n)}= \sum_{k=0}^{{\rm min}\{ m,n\}}

Theorems & Definitions (20)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 4.1
  • Lemma 4.2
  • ...and 10 more