Quantization of Lie bialgebras, I

Selecta Mathematica, 2000

This paper is a continuation of . In [EK3], we introduced the Hopf algebra F (R) z associated to a quantum R-matrix R(z) with a spectral parameter defined on a 1-dimensional connected algebraic group Σ, and a set of points z = (z 1 , . . . , z n ) ∈ Σ n . This algebra is generated by entries of a matrix power series T i (u), i = 1, . . . , n, subject to Faddeev-Reshetikhin-Takhtajan type commutation relations, and is a quantization of the group GL N [[t]] n .

Transformation Groups - TRANSFORM GROUPS, 2008

This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid...

Selecta Mathematica-new Series - SEL MATH-NEW SER, 2000

This paper is a continuation of [EK1]{[EK4]. The goal here is to dene and study the notion of a quantum vertex operator algebra (VOA) in the setting of the formal deformation theory and give interesting examples of such algebras. Our denition of a quantum VOA is based on the ideas of the paper [FrR]. The rst section of our paper is devoted to the general theory of quantum VOAs. For simplicity, we consider only bosonic algebras, but all the denitions and results admit a straightforward generalization to the supercase. We start with the version of the denition of a VOA in which the main axiom is the locality (commutativity) axiom. To obtain a quantum deformation of this denition, we replace the locality axiom with the S-locality axiom, whereS is a shift-invariant unitary solution of the quantum Yang-Baxter equation (the other axioms are unchanged). We call the obtained structure a braided VOA. However, a braided VOA does not necessarily satisfy the associativity property, which is one...

Communications in Mathematical Physics, 2008

We study classical twists of Lie bialgebra structures on the polynomial current algebra g [u], where g is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called quasi-trigonometric solutions of the classical Yang-Baxter equation. It turns out that quasi-trigonometric r-matrices fall into classes labelled by the vertices of the extended Dynkin diagram of g. We give complete classification of quasi-trigonometric r-matrices belonging to multiplicity free simple roots (which have coefficient 1 in the decomposition of the maximal root). We quantize solutions corresponding to the first root of sl(n).

2002

We quantize the Poisson-Lie group SL(2,R)^* as a bialgebra using the product of Kontsevich. The coproduct is a deformation of the coproduct that comes from the group structure. The resulting bialgebra structure is isomorphic to the quantum universal enveloping algebra U_hsl(2,R).

Journal of Mathematical Physics, 2000

We construct a deformation of the function algebra on the quantum group SL q (2) into a trialgebra in the sense of Crane and Frenkel. We show that this naturally acts on the trialgebraic deformation of the Manin plane, previously introduced by the authors. Alternatively, one can view it as acting on the trialgebraic deformation of the fermionic Manin plane. We prove that the trialgebraic deformation of SL q (2) defines a 2 − C *-category, a structure as needed for the superselection structure of massive two dimensional quantum field theories. Besides this, we investigate another approach to trialgebra deformations of a bialgebra as a deformation of a Fock space construction over the bialgebra.

Frontiers of Mathematics in China, 2010

In this paper, we use the general quantization method by Drinfel'd twists to quantize the Schrödinger-Virasoro Lie algebra whose Lie bialgebra structures were recently discovered by Han-Li-Su. We give two different kinds of Drinfel'd twists, which are then used to construct the corresponding Hopf algebraic structrues. Our results extend the class of examples of noncommutative and noncocommutative Hopf algebras.

Journal of Mathematical Sciences, 2005

We perform a simultaneous multiparameter quantization of some three-dimensional Lie algebras, such as the Heisention is possible since any of the mentioned algebras is dual to the same solvable Lie algebra. We present an explicit form of the numerical R-matrix; this allows us to represent some of the commutation relations in the form of the P T T-equation. Bibliography: 14 titles.

Arxiv preprint q-alg/9605026, 1996

1996

Let $ G^\tau $ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let $ H^\tau $ be its dual Poisson group. By means of Drinfeld's double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,\phi}^M ({\frak h}) $ --- dual of $ U_{q,\phi}^{M'} ({\frak g}) $ --- which yield infinitesimal quantization of $ H^\tau $ and $ G^\tau $; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $ H^\tau $ what was known for $ G^\tau $, completing the quantization of the pair $ (G^\tau,H^\tau) $.

Communications in Mathematical Physics, 1999

Let A be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, G, with the Lie algebra g. We study one and two parameter quantizations A h and A t,h of A such that the multiplication on the quantized algebra is invariant under action of the Drinfeld-Jimbo quantum group, U h (g).

Pacific Journal of Mathematics, 1998

Let G τ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group; let H τ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups U M q,φ (h)-dual of U M ′ q,φ (g)-which yield infinitesimal quantization of H τ and G τ ; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for H τ what was known for G τ , completing the quantization of the pair (G τ , H τ).

Journal of Mathematical Physics, 1995

A Poisson-Hopf algebra of smooth functions is simultaneously constructed on the two dimensional Euclidean, Poincare, and Heisenberg groups by using a classical r-matrix which is invariant under contraction. The quantization for this algebra of functions is developed, and its dual Hopf algebra is also computed. Contractions on these quantum groups are studied. It is shown that, within this setting, classical deformations are transformed into quantum ones by Hopf algebra duality and the quantum Heisenberg algebra is derived by means of a (dual) Poisson-Lie quantization that deforms the standard Moyal-Weyl ah-product. 0 1995 American Institute of Physics.

2010

We apply the star product quantization to the Lie algebra. The quantization in terms of the star product is well known and the commutation relation in this case is called the θ-deformation where the constant θ appears as a parameter. In the application to the Lie algebra, we need to change the parameter θ to x-dependent θ(x). There is no essential difference between the quantization in the quantum mechanics and deriving quantum numbers in the Lie algebra from the viewpoint of the star product. We propose to unify them in higher dimensions, which may be analogous to the Kaluza-Klein theory in the classical theory.

1999

All Lie bialgebra structures for the (1+1)-dimensional centrally extended Schrodinger algebra are explicitly derived and proved to be of the coboundary type. Therefore, since all of them come from a classical r-matrix, the complete family of Schrodinger Poisson-Lie groups can be deduced by means of the Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended Galilei and gl(2) Lie bialgebras within the Schrodinger classification are studied. As an application, new quantum (Hopf algebra) deformations of the Schrodinger algebra, including their corresponding quantum universal R-matrices, are constructed.