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

TL;DR

This paper characterizes the structure of projective indecomposable modules for a specific subalgebra of the hyperalgebra of SL_2, utilizing primitive idempotents to deepen understanding of module theory in this context.

Contribution

It provides a detailed description of projective modules for the degree 0 subalgebra in the hyperalgebra of SL_2, introducing new structural insights and methods.

Findings

Explicit structure of projective indecomposable modules identified

Primitive idempotents used to analyze module decomposition

Enhanced understanding of hyperalgebra representations for SL_2

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.