Triangular matrix algebras over affine quasi-hereditary algebras (Q2666912)

Quasi-hereditary algebras (and its module categories called highest weight categories) were introduced by \textit{E. Cline} et al. [J. Reine Angew. Math. 391, 85--99 (1988; Zbl 0657.18005)] with many applications to representation theory and other related fields. There are many ways to construct new quasi-hereditary algebras from old ones. Affine quasi-hereditary algebras were introduced recently by \textit{A. S. Kleshchev} [Proc. Lond. Math. Soc. (3) 110, No. 4, 841--882 (2015; Zbl 1360.16010)] which generalize quasi-hereditary algebras. The author of the paper under review proves that triangular matrix algebras over affine quasi-hereditary algebras are affine quasi-hereditary algebras, which generalizes the result ``the triangular matrix algebras over quasi-hereditary algebras are also quasi-hereditary'' proved by the reviewer, see [Tsukuba J. Math. 25, No. 1, 1--11 (2001; Zbl 1036.16009)]. The main result of the paper is the following: Theorem (Theorem 4.1.). Let \(H\) be an affine quasi-hereditary algebra, and \(T_2(H)=\left(\begin{array}{cc}H&0\\ H&H\end{array}\right)\). Then \(T_2(H)\) is an affine quasi-hereditary algebra if and only if H is an affine quasi-hereditary algebra. The author also gives a description of tilting module and global dimension of \(T_2(H)\). In the appendix, it is proven the affine quasi-heredity of some centralizer algebras of an affine quasi-hereditary algebra.