Searcharxiv⌕ Search

arXiv subjects

Nicolai Reshetikhin

Publications and source records attributed to Nicolai Reshetikhin.

58 records · Page 4Linked to original sources
arXiv1997

Quantization of the Algebra of Chord Diagrams

In this paper we define an algebra structure on the vector space $L(Σ)$ generated by links in the manifold $Σ\times [0,1]$ where $Σ$ is an oriented surface. This algebra has a filtration and the associated graded algebra $L_{Gr}(Σ)$ is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams $ch(Σ)$ on $Σ$ to $L_{Gr}(Σ)$. We show that multiplication in $L(Σ)$ provides a geometric way to define a deformation quantization of the algebra of chord diagrams, provided there is a universal Vassiliev invariant for links in $Σ\times [0,1]$. The quantization descends to a quantization of the moduli space of flat connections on $Σ$ and it is universal with respect to group homomorphisms. If $Σ$ is compact with free fundamental group we construct a universal Vassiliev invariant.

q-alg↗
arXiv1996

Semiquantum Geometry

In this paper we study associative algebras with a Poisson algebra structure on the center acting by derivations on the rest of the algebra. These structures, which we call Poisson fibred algebras, appear in the study of quantum groups at roots of 1 and related algebras, as well as in the representation theory of affine Lie algebras at the critical level. Poisson fibred algebras lead to a generalization of Poisson geometry, which we develop in the paper. We also take up the general study of noncommutative spaces which are close to enough commutative ones so that they contain enough points to have interesting commutative geometry. One of the most striking uses of our noncommutative spaces is the quantum Borel-Weil-Bott Theorem for quantum sl_q (2) at a root of unity, which comes as a calculation of the cohomology of actual sheaves on actual topological spaces.

q-alg↗
arXiv1994

Quasitriangularity of quantum groups at roots of 1

An important property of a Hopf algebra is its quasitriangularity and it is useful various applications. This property is investigated for quantum groups $sl_2$ at roots of 1. It is shown that different forms of the quantum group $sl_2$ at roots of 1 are either quasitriangular or have similar structure which will be called autoquasitriangularity. In the most interesting cases this property means that "braiding automorphism" is a combination of some Poisson transformation and an adjoint transformation with certain element of the tensor square of the algebra.

hep-th↗