Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuantum deformations of projective three-space
246
0
0.0
(
0
)
Authors
Brent Pym
Ask ChatGPT about the research
No Arabic abstract
We describe the possible noncommutative deformations of complex projective three-space by exhibiting the Calabi--Yau algebras that serve as their homogeneous coordinate rings. We prove that the space parametrizing such deformations has exactly six irreducible components, and we give explicit presentations for the generic members of each family in terms of generators and relations. The proof uses deformation quantization to reduce the problem to a similar classification of unimodular quadratic Poisson structures in four dimensions, which we extract from Cerveau and Lins Netos classification of degree-two foliations on projective space. Corresponding to the ``exceptional component in their classification is a quantization of the third symmetric power of the projective line that supports bimodule quantizations of the classical Schwarzenberger bundles.
rate research
Read More
We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and Jordanian types
. A part of twists corresponding to the r-matrices of Zakrzewski classification are given in explicit form.
We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated r-matrices of Abelian and Jordanian types. Corresponding twists describing quantum deformations are obtained in explicit form. This work is an extended version of the paper url{arXiv:0704.0081v1 [math.QA]}.
We discuss quantum deformations of Jordanian type for Lie superalgebras. These deformations are described by twisting functions with support from Borel subalgebras and they are multiparameter in the general case. The total twists are presented in explicit form for the Lie superalgebras sl(m|n) and osp(1|2n). We show also that the classical $r$-matrix for a light-cone deformation of D=4 super-Poincare algebra is of Jordanian type and a corresponding twist is given in explicit form.
We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking the spectra of quantum g^-KdV Hamiltonians and affine opers for the Langlands dual affine algebra. As a bonus, we obtain a direct and uniform proof of the Bethe Ansatz equations for a large class of quantum integrable models associated to arbitrary untwisted quantum affine algebras, under a mild genericity condition. We also conjecture analogues of these results for the twisted quantum affine algebras and elucidate the notion of opers for twisted affine algebras, making a connection to twisted opers introduced in [FG].
The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
