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Projective modules for the subalgebra of degree 0 in a finite-dimensional hyperalgebra of type $A_1$

Yutaka Yoshii

Abstract

We describe the structure of projective indecomposable modules for the subalgebra consisting of the elements of degree 0 in the hyperalgebra of the $r$-th Frobenius kernel for the algebraic group ${\rm SL}_2(k)$, using the primitive idempotents which were constructed before by the author.

Projective modules for the subalgebra of degree 0 in a finite-dimensional hyperalgebra of type $A_1$

Abstract

We describe the structure of projective indecomposable modules for the subalgebra consisting of the elements of degree 0 in the hyperalgebra of the -th Frobenius kernel for the algebraic group , using the primitive idempotents which were constructed before by the author.

Paper Structure

This paper contains 3 sections, 13 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PIMs for $\mathcal{A}_r$

Key Result

Lemma 3.1

Let $(a,j) \in \mathcal{P}$. Then the following holds. (i) $n^{(0)}(a,j)= (p-a-1)/2 +j\ \hbox{and} \ n^{(1)}(a,j)= (3p-a-1)/2 -j$ under (A), (ii) $n^{(0)}(a,j)= (p-a-1)/2 -j\ \hbox{and} \ n^{(1)}(a,j)= (p-a-1)/2 +j$ under (B), (iii) $n^{(0)}(a,j)= (2p-a-1)/2 -j \ \hbox{and} \ n^{(1)}(a,j)= (2p-a-1

Theorems & Definitions (13)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Theorem 3.8
  • Lemma 3.9
  • Proposition 3.10
  • ...and 3 more