97 research outputs found
Axiomatic framework for the BGG Category O
We introduce a general axiomatic framework for algebras with triangular
decomposition, which allows for a systematic study of the
Bernstein-Gelfand-Gelfand Category . The framework is stated via
three relatively simple axioms; algebras satisfying them are termed "regular
triangular algebras (RTAs)". These encompass a large class of algebras in the
literature, including (a) generalized Weyl algebras, (b) symmetrizable
Kac-Moody Lie algebras , (c) quantum groups
over "lattices with possible torsion", (d) infinitesimal Hecke algebras, (e)
higher rank Virasoro algebras, and others.
In order to incorporate these special cases under a common setting, our
theory distinguishes between roots and weights, and does not require the Cartan
subalgebra to be a Hopf algebra. We also allow RTAs to have roots in arbitrary
monoids rather than root lattices, and the roots of the Borel subalgebras to
lie in cones with respect to a strict subalgebra of the Cartan subalgebra.
These relaxations of the triangular structure have not been explored in the
literature.
We then study the BGG Category over an RTA. In order to work
with general RTAs - and also bypass the use of central characters - we
introduce conditions termed the "Conditions (S)", under which distinguished
subcategories of Category possess desirable homological
properties, including: (a) being a finite length, abelian, self-dual category;
(b) having enough projectives/injectives; or (c) being a highest weight
category satisfying BGG Reciprocity. We discuss whether the above examples
satisfy the various Conditions (S). We also discuss two new examples of RTAs
that cannot be studied using previous theories of Category , but
require the full scope of our framework. These include the first construction
of algebras for which the "root lattice" is non-abelian.Comment: 59 pages, LaTeX. This paper supersedes (and goes far beyond) the
older preprint arXiv:0811.2080 (31 pages), which has been completely
rewritte
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