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On the Kazhdan-Lusztig theory of dual extension quasi-hereditary algebras. (English) Zbl 1212.16038

Summary: In order to study the representation theory of Lie algebras and algebraic groups, E. Cline, B. Parshall and L. Scott [TĂ´hoku Math. J., II. Ser. 45, No. 4, 511-534 (1993; Zbl 0801.20013)] introduced the notion of abstract Kazhdan-Lusztig theory for quasi-hereditary algebras. Assume that a quasi-hereditary algebra \(B\) has the vertex set \(Q_0=\{1,\dots,n\}\) such that \(\operatorname{Hom}_B(P(i),P(j))=0\) for \(i>j\). In this paper, it is shown that if the quasi-hereditary algebra \(B\) has a Kazhdan-Lusztig theory relative to a length function \(l\), then its dual extension algebra \(A=\mathcal A(B)\) has also the Kazhdan-Lusztig theory relative to the length function \(l\).

MSC:

16G20 Representations of quivers and partially ordered sets
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras

Citations:

Zbl 0801.20013
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