The Kostant form of $\mathfrak{U}(sl_n^+)$ and the Borel subalgebra of the Schur algebra S(n,r)

TL;DR

This paper constructs a functor linking graded modules over the Kostant form of the positive part of the universal enveloping algebra of sl_n to modules over the Borel subalgebra of the Schur algebra, preserving projective resolutions.

Contribution

It introduces a new functor that connects graded modules over the Kostant form to modules over the Schur algebra's Borel subalgebra, preserving minimal projective resolutions.

Findings

The functor maps projective resolutions of one-dimensional modules to those of simple modules.

Establishes a correspondence between graded modules over $A_n(K)$ and modules over $S^+(n,r)$.

Provides a new tool for studying representations of Schur algebras via Kostant forms.

Abstract

Let $A_n(K)$ be the Kostant form of $\mathfrak{U}(sl_n^+)$ and $\Gamma$ the monoid generated by the positive roots of $sl_n$. For each $\lambda\in \Lambda(n,r)$ we construct a functor $F_{\lambda}$ from the category of finitely generated $\Gamma$-graded $A_n(K)$-modules to the category of finite dimensional $S^+(n,r)$-modules, with the property that $F_{\lambda}$ maps (minimal) projective resolutions of the one-dimensional $A_n(K)$-module $K_{A}$ to (minimal) projective resolutions of the simple $S^+(n,r)$-module $K_{\lambda}$.