Presenting quantum Schur algebras as quotients of the quantized universal
enveloping algebra of $\mathfrak{gl}_2$
Stephen Doty and Anthony Giaquinto

Abstract. We obtain a presentation of the quantum Schur algebras $S_v(2,d)$
by generators and relations. This presentation is compatible with the
usual presentation of the quantized enveloping algebra $\mathbf{U} = 
\mathbf{U}_v(\mathfrak{gl}_2)$. In the process we find new bases for 
$S_v(2,d)$. We also locate the $\mathbb Z[v,v^{-1}]$-form of the quantum 
Schur algebra with the presented algebra and show that it has a basis which 
is closely related to Lusztig's basis of teh $\mathbb Z[v,v^{-1}]$-form of 
$\mathbf{U}$.