Linear quivers and the geometric setting of quantum \(\mathrm{GL}_n\).
Let \({\mathcal Z}=\mathbb Z[\nu,\nu^{-1}]\) and \(\Delta=(\Delta_0,\Delta_1)\) a quiver such that the underlying graph is a simply-laced connected Dynkin graph. Let \(\Phi^+\) be the set of positive roots and \(\Theta=\{f\colon\Phi^+\to\mathbb N\}\). The twisted generic Hall algebra \(H(\Delta)\) over \(\mathcal Z\) has a \(\mathcal Z\)-free basis \(\{u_f\}_{f\in\Theta}\) and the product is defined in terms of Hall polynomials and the Euler form. \textit{C. M. Ringel} [Aportaciones Mat., Comun. 15, 85--114 (1995)] proved that \(H(\Delta)\) is isomorphic to the Lusztig form (a \(\mathcal Z\)-subalgebra) of the positive part of the quantized enveloping algebra over \(\mathbb Q(\nu)\) associated to the semisimple Lie algebra corresponding to \(\Delta\). When \(\Delta\) is a linear quiver the authors provide better relations for the product in \(H(\Delta)\). There is an algebra isomorphism between \(\mathbb H(\Delta)=H(\Delta)\otimes_{\mathcal Z}\mathbb Q(\nu)\) and the possitive part \(\mathbb V^+\) of the BLM algebra defined by \textit{A. A. Beilinson, G. Lusztig} and \textit{R. MacPherson} [Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)]. The authors extend also this isomorphism to \(H(\Delta)\) and a suitable \(\mathcal Z\)-subalgebra \(V^+\). This allows the computation of a PBW basis for \(V^+\). At this point the authors present two posets defined in terms of some algebraic varieties; the first one is the opposite of the degeneration order on \(\Phi^+\) defined by \textit{K. Bongartz} [Adv. Math. 121, No. 2, 245-.287 (1996; Zbl 0862.16007)], and the second one is the Bruhat order on some matrices with non-negative entries. Both orders are compared. The last family of algebras covered by the authors are the \(q\)-Schur algebras \(\mathbb U_r\). The authors remember a basis of \(\mathbb U_r\) coming from BLM algebras and they extend it to a \(\mathcal Z\)-subalgebra denoted by \(U_r\). In the next section the kernel of the canonical epimorphism from the Hall algebra \(H\) to the positive part of the integral \(q\)-Schur algebra \(U_r\) is computed in terms of distinguished words in the set \(\Delta_0\). This epimorphism is defined via the identification of \(H\) with the positive part of the integral quantized enveloping algebra \(U\). Finally, the isomorphism between the Hall algebra and the positive part \(V^+\) of the BLM algebra previously proven allows the computation of new bases for the nonnegative and nonpositive parts of the \(q\)-Schur algebras.
scientific article; zbMATH DE number 1961120
scientific article