2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167868It 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 $G$.18 pages; to appear in J. AlgebraQuantum AlgebraRings and Algebrastext