Open AccessKoszul duality for parabolic and singular category đť’Ş
Author(s) -
Erik Backelin
Publication year - 1999
Publication title -
representation theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.169
H-Index - 37
ISSN - 1088-4165
DOI - 10.1090/s1088-4165-99-00055-2
Subject(s) - mathematics , duality (order theory) , type (biology) , algorithm , block (permutation group theory) , algebra over a field , combinatorics , pure mathematics , ecology , biology
This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category O \mathcal O over a complex semi-simple Lie-algebra, established by Beilinson, Ginzburg and Soergel in Koszul duality patterns in representation theory , J. Amer. Math. Soc. 9 (1996), 473–527. In that paper it was proved that any “block” in O \mathcal O , determined by an integral, but possibly singular weight, is Koszul (i.e. equivalent to the category of finitely generated modules over some Koszul ring) and, moreover, that the “Koszul dual” of such a block is isomorphic to a “parabolic subcategory” of the trivial block in O \mathcal O . We extend these results to prove that a parabolic subcategory of an integral and (possibly) singular block in O \mathcal O is Koszul and we also calculate the Koszul dual of such a category.