\(q\)-Schur algebras as quotients of quantized enveloping algebras
Let \({\mathcal A}: = \mathbb{Z} [v,v^{-1}]\). Let \(U(gl_n)\) be the quantized enveloping algebra over \(\mathbb{Q} (v)\) corresponding to the Lie algebra \(gl_n\), and \(U_{\mathcal A} (gl_n)\) its \({\mathcal A}\) form. Let \(S_q(n,r)\) be the \(q\)-Schur algebra, which is an associative algebra with 1 with base ring \({\mathcal A}\) (the relationship between \(q\) and \(v\) is given by \(q= v^2)\). It is known that there exists a surjective algebra homomorphism [\textit{A. A. Beilinson}, \textit{G. Lusztig} and \textit{R. MacPherson}, Duke Math. J. 61, 655-677 (1990; Zbl 0713.17012)] \[ \theta_r: U_{\mathcal A} (gl_n) \to S_v (n,r). \] In this paper the author studies the properties of the surjective algebra homomorphism, \(\theta = \theta_r\). He finds preimages of elements of the \(q\)-Schur algebra under this map and gives a description of \(\ker \theta\).
scientific article; zbMATH DE number 954778