Ringel duals of affine quasi-hereditary algebras (Q2236078)

The class of affine quasi-hereditary algebras was introduced by \textit{A. S. Kleshchev} [Proc. Lond. Math. Soc. (3) 110, No. 4, 841--882 (2015; Zbl 1360.16010)]. Note that it is a graded version of the class of quasi-hereditary algebras introduced by \textit{E. Cline} et al. [J. Reine Angew. Math. 391, 85--99 (1988; Zbl 0657.18005)]. \textit{C. M. Ringel} [Math. Z. 208, No. 2, 209--224 (1992; Zbl 0725.16011)] studied the endomorphism algebra \(\mathcal{R}\) of the characteristic tilting module of a quasi-hereditary algebra \(H\), which is now called the Ringel dual of \(H\). He proved that this endomorphism algebra is a quasi-hereditary algebra. In the paper under review, the author extends Ringel's result to the case when \(H\) is an affine quasi-hereditary algebra finitely generated over its center and satisfying some additional condition. Moreover, the author considers the double Ringel dual \(\mathcal{RR}\) of \(H\) and proves (under the same conditions) that \(\mathcal{RR}\) is graded Morita equivalent to \(H\). In particular, if all the irreducible \(H\)-modules have dimension one, then \(\mathcal{RR} \cong H\) as graded algebras. Finally, let us mention that the theory of affine quasi-hereditary algebras has important applications in the theory of representations of many classical infinite dimensional algebras such as KLR algebras and affine Hecke algebras.