On the Kazhdan-Lusztig Theory of Dual Extension Quasi-Hereditary Algebras

Abstract:

      In order to study the representation theory of Lie algebras and algebraic groups, Cline, Parshall and Scott put forward the notion of abstract Kazhdan-Lusztig theory for quasi-hereditary algebras. Assume that a quasi-hereditary algebra $B$ has the vertex set $Q_0=\{1, \ldots, n\}$ such that 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 $A={\cal A}(B)$ has also the Kazhdan-Lusztig theory relative to the length function $l$.