Exact Borel subalgebras of quasi-hereditary algebras. I. (With an appendix by Leonard Scott)
type: Item
instance of: scholarly article
title: Exact Borel subalgebras of quasi-hereditary algebras. I. (With an appendix by Leonard Scott)
zbMATH Open document ID: 0841.16013
DOI: 10.1007/BF02572622
published in: Mathematische Zeitschrift
publication date: 1996-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174817
review text: 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.
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 856572
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: complex semisimple Lie algebras
zbMATH Keywords: induction functor
zbMATH Keywords: simple \(B\)-modules
zbMATH Keywords: Weyl modules
zbMATH Keywords: exact Borel subalgebras
zbMATH Keywords: composition multiplicities
zbMATH Keywords: hereditary ideals
zbMATH Keywords: indecomposable projective modules
zbMATH Keywords: blocks of category \(\mathcal O\)
zbMATH Keywords: generalized Schur algebras
zbMATH Keywords: polynomial representation
zbMATH Keywords: semisimple algebraic groups
author: Steffen Koenig
reviewed by: Steffen Koenig
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 856572