20 research outputs found
On the quantization of Poisson brackets
In this paper we introduce two classes of Poisson brackets on algebras (or on
sheaves of algebras). We call them locally free and nonsingular Poisson
brackets. Using the Fedosov's method we prove that any locally free nonsingular
Poisson bracket can be quantized. In particular, it follows from this that all
Poisson brackets on an arbitrary field of characteristic zero can be quantized.
The well known theorem about the quantization of nondegenerate Poisson brackets
on smooth manifolds follows from the main result of this paper as well.Comment: Latex, 24 pp., essentially corrected versio
Quantum symmetric pairs and representations of double affine Hecke algebras of type
We build representations of the affine and double affine braid groups and
Hecke algebras of type , based upon the theory of quantum symmetric
pairs . In the case , our constructions provide a
quantization of the representations constructed by Etingof, Freund and Ma in
arXiv:0801.1530, and also a type generalization of the results in
arXiv:0805.2766.Comment: Final version, to appear in Selecta Mathematic
Quantized multiplicative quiver varieties
Beginning with the data of a quiver Q, and its dimension vector d, we
construct an algebra D_q=D_q(Mat_d(Q)), which is a flat q-deformation of the
algebra of differential operators on the affine space Mat_d(Q). The algebra D_q
is equivariant for an action by a product of quantum general linear groups,
acting by conjugation at each vertex. We construct a quantum moment map for
this action, and subsequently define the Hamiltonian reduction A^lambda_d(Q) of
D_q with moment parameter \lambda. We show that A^\lambda_d(Q) is a flat formal
deformation of Lusztig's quiver varieties, and their multiplicative
counterparts, for all dimension vectors satisfying a flatness condition of
Crawley-Boevey: indeed the product on A^\lambda_d(Q) yields a Fedosov
quantization the of symplectic structure on multiplicative quiver varieties. As
an application, we give a description of the category of representations of the
spherical double affine Hecke algebra of type A_{n-1}, and its generalization
constructed by Etingof, Oblomkov, and Rains, in terms of a quotient of the
category of equivariant D_q-modules by a Serre sub-category of aspherical
modules.Comment: Re-written introduction, improvements to expositio