17 research outputs found
Quantum flag varieties, equivariant quantum D-modules and localization of quantum groups
Let \Oq(G) be the algebra of quantized functions on an algebraic group
and \Oq(B) its quotient algebra corresponding to a Borel subgroup of .
We define the category of sheaves on the "quantum flag variety of " to be
the \Oq(B)-equivariant \Oq(G)-modules and proves that this is a
proj-category. We construct a category of equivariant quantum
-modules on this quantized flag variety and prove the
Beilinson-Bernsteins localization theorem for this category in the case when
is not a root of unity
Higher Auslander-Reiten sequences and -structures
Let be an artin algebra and an additive subcategory of
. We construct a -structure on the homotopy category
whose heart is
a natural domain for higher Auslander-Reiten (AR) theory. The abelian
categories (which is the natural domain
for classical AR theory) and interact via various
functors. If is functorially finite then
is a quotient category of
. We illustrate the theory with two
examples:
Iyama developed a higher AR theory when is a maximal
-orthogonal subcategory, see \cite{I}. In this case we show that the simple
objects of correspond to Iyama's higher AR
sequences and derive his higher AR duality from the existence of a Serre
functor on the derived category
.
The category of a complex semi-simple Lie algebra
fits into higher AR theory by considering to be the
coinvariant algebra of the Weyl group of .Comment: 26 pages, accepted for publication in Journal of Algebra 201
Global quantum differential operators on quantum flag manifolds, theorems of Duflo and Kostant
We give a short new proof for the theorem that global sections of the sheaf
of quantum differential operators on a quantum flag manifold are given by the
quantum group. As corollaries we retrieve Joseph and Letzter's quantum versions
of classical enveloping algebra theorems of Duflo and Kostant. We also describe
the center of the ad-integrable part of the quantum group and the adjoint Lie
algebra action on it.Comment: 14 pages, fixed some error
Projective and Whittaker functors on category
We show that the Whittaker functor on a regular block of the BGG-category
of a semisimple complex Lie algebra can be obtained by composing
a translation to the wall functor with Soergel and Mili\v{c}i\'{c}'s
equivalence between the category of Whittaker modules and a singular block of
. We show that the Whittaker functor is a quotient functor that
commutes with all projective functors and endomorphisms between them.Comment: 14 page
Endomorphisms of quantized Weyl algebras
Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the
n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are
canonically isomorphic. We discuss how this conjecture can be approached by
means of (second) quantized Weyl algebras at roots of unity
Koszul duality for parabolic and singular category
Abstract. This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category O over a complex semisimple 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, 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. We extend these results to prove that a parabolic subcategory of an integral and (possibly) singular block in O is Koszul and we also calculate the Koszul dual of such a category. 1