Double centralizer properties, dominant dimension, and tilting modules

Citation
S. Konig et al., Double centralizer properties, dominant dimension, and tilting modules, J ALGEBRA, 240(1), 2001, pp. 393-412
Citations number
23
Language
INGLESE
art.tipo
Article
Categorie Soggetti
Mathematics
Journal title
JOURNAL OF ALGEBRA
ISSN journal
0021-8693 → ACNP
Volume
240
Issue
1
Year of publication
2001
Pages
393 - 412
Database
ISI
SICI code
0021-8693(20010601)240:1<393:DCPDDA>2.0.ZU;2-K
Abstract
Double centralizer properties play a central role in many parts of algebrai c Lie theory. Soergel's double centralizer theorem relates the principal bl ock of the Bernstein-Gelfand-Gelfand category O of a semisimple complex Lie algebra with the coinvariant algebra (i.e., the cohomology algebra of the corresponding flag manifold). Schur-Weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic , or (via the quantized version) in nondefining characteristic. In this pap er we exhibit algebraic structures behind these double centralizer properti es. We show that the finite dimensional algebras relevant in this context h ave dominant dimension at least two with respect to some projective-injecti ve or tilting modules. General arguments which combine methods from ring th eory (QF-3 rings and dominant dimension) with tools from representation the ory (approximations, tilting modules) then imply the validity of these doub le centralizer properties as well as new ones. In contrast to the tradition al proofs (e.g., by the fundamental theorems of invariant theory) no comput ations are necessary. (C) 2001 Academic Press.