Exact Borel subalgebras of quasi-hereditary algebras. I. (With an appendix by Leonard Scott)
Exact Borel subalgebras of quasi-hereditary algebras are defined in analogy to some features of representation theory of complex semisimple Lie algebras: An exact Borel subalgebra \(B\) of a quasi-hereditary algebra \((A, \leq)\) comes with an exact induction functor sending the simple \(B\)-modules to the Weyl modules of \((A, \leq)\), and \((B, \leq)\) is directed, hence looks like coming from a solvable Lie subalgebra. Several properties of exact Borel subalgebras are studied. In particular, exact Borel subalgebras are characterized in terms of composition multiplicities and intersections with hereditary ideals. Moreover, a close connection is described between exact Borel subalgebras and what is called \(\Delta\)-subalgebras, whose indecomposable projective modules are natural representatives of Weyl modules over the opposite algebra \((A^{\text{op}}, \leq)\). Existence of exact Borel subalgebras is proved for certain classes of quasi-hereditary algebras. The main result is existence of exact Borel subalgebras for the quasi-hereditary algebras to blocks of category \(\mathcal O\) which have been constructed by Bernstein, Gelfand and Gelfand. The main result of the appendix is the existence of exact Borel subalgebras for Morita-equivalent versions of the generalized Schur algebras which describe polynomial representation theory of semisimple algebraic groups.
scientific article; zbMATH DE number 856572
scientific article