Little \(q\)-Schur algebras at even roots of unity
In joint work with \textit{J. Du} and \textit{J.-P. Wang} [J. Algebra 287, No. 1, 199--233 (2005; Zbl 1140.17301)] the author of the paper under review introduced little \(q\)-Schur algebras as homomorphic images of Lusztig's infinitesimal quantum \(\mathfrak{gl}_n\) for primitive roots of unity of odd order in a field \(k\) of characteristic zero. In this paper the author extends the definition of little \(q\)-Schur algebras to primitive roots of unity of any order \(l^\prime\geq 3\) by using a certain subalgebra \(\widetilde{u}_k(n)\) of the quantized hyperalgebra of \(\mathfrak{gl}_n\) instead of Lusztig's infinitesimal quantum \(\mathfrak{gl}_n\). More importantly, the author also defines an algebra \(\mathcal{W}\) coming from the geometric realization of the quantized universal enveloping algebra of \(\mathfrak{gl}_n\) due to \textit{A. A. Beilinson}, \textit{G. Lusztig}, and \textit{R. MacPherson} [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)] for even roots of unity and proves that there is an algebra epimorphism \(\varphi\) from \(\widetilde{u}_k(n)\) onto \(\mathcal{W}\) which is an isomorphism if \(l^\prime\) is even and otherwise \(\widetilde{u}_k (n)/\text{Ker}(\varphi)\cong \mathcal{W}\) is Lusztig's infinitesimal quantum \(\mathfrak{gl}_n\) so that in both cases little \(q\)-Schur algebras can be obtained as homomorphic images of \(\mathcal{W}\). By using the geometric setting the author then constructs various bases for the little \(q\)-Schur algebra similar to the case of an odd root of unity and as a consequence he derives dimension formulas for the little \(q\)-Schur algebra and its zero- and positive parts.
scientific article; zbMATH DE number 5148860
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