Strong exact Borel subalgebras of quasi-hereditary algebras and abstract Kazhdan-Lusztig theory
The author continues to study the Borel subalgebras of quasi-hereditary algebras. The notion of Borel subalgebras was introduced by the author in his earlier papers. The main problem studied in this paper is the following question: suppose there is a Borel subalgebra \(B\) for a quasi-hereditary algebra \(A\). Does \(A\) possess a strong exact Borel subalgebra? Here a strong Borel subalgebra \(B\) is by definition a Borel subalgebra of \(A\) such that \(B\) and \(A\) have the same maximal semisimple subalgebra. To this problem the author gives an affirmative answer, and he also deduces several consequences. The precise formulations of the main results are the following: (1) Let \((A,\leq)\) be a quasi-hereditary algebra and \(B\) an exact Borel subalgebra of \((A,\leq)\) such that \(B\) is basic. Suppose \(S\) is a maximal semisimple subalgebra of \(B\). Then there is a maximal subalgebra \(S(A)\) of \(A\) which contains \(S\) such that the subalgebra of \(A\) generated by \(B\) and \(S(A)\) is a strong exact Borel subalgebra of \((A,\leq)\). There is a similar statement for strong \(\Delta\)-subalgebras. (2) Let \((A,\leq)\) be a quasi-hereditary algebra and \((A',\leq)\) a quasi-hereditary algebra which is Morita equivalent to \((A,\leq)\). If \((A,\leq)\) has an exact Borel subalgebra, then so does \((A',\leq)\). There is a similar statement for \(\Delta\)-subalgebras. (3) Let \(\mathfrak g\) be a finite dimensional semisimple complex Lie algebra. Let \(A\) be an algebra associated with a block of category \(\mathcal O\) of \(\mathfrak g\). Then the quasi-hereditary algebra \(A\) has a strong exact Borel subalgebra \(B\) and a \(\Delta\)-subalgebra \(C\) such that \(S=B\cap C\) is a maximal semisimple subalgebra of \(A\). The paper ends with some examples which illustrate the results and the notions in the paper as well as the relationships of representation types and \(\Delta\)-good modules between the algebras and their Borel subalgebras.
scientific article; zbMATH DE number 1389128
scientific article