Constructing quantized enveloping algebras via inverse limits of finite dimensional algebras

TL;DR

The paper demonstrates that quantized enveloping algebras and their modified forms can be constructed as inverse limits of generalized q-Schur algebras, clarifying their relationship and providing a new perspective on their structure.

Contribution

It introduces a novel construction of quantized enveloping algebras via inverse limits of finite-dimensional algebras, enhancing understanding of their structure and relations.

Findings

Constructed $ ext{U}$ and $ ext{U}^ullet$ as inverse limits of generalized q-Schur algebras.

Clarified the relation between $ ext{U}$ and $ ext{U}^ullet$ using inverse limit framework.

Identified the inverse limit as a q-analogue of the dual of the coordinate algebra of a linear algebraic group.

Abstract

It is known that a generalized $q$-Schur algebra may be constructed as a quotient of a quantized enveloping algebra $\UU$ or its modified form $\dot{\UU}$. On the other hand, we show here that both $\UU$ and $\dot{\UU}$ may be constructed within an inverse limit of a certain inverse system of generalized $q$-Schur algebras. Working within the inverse limit $\hat{\UU}$ clarifies the relation between $\dot{\UU}$ and $\UU$. This inverse limit is a $q$-analogue of the linear dual $R[G]^*$ of the coordinate algebra of a corresponding linear algebraic group $G$.