Basic Twist Quantization of the Exceptional Lie Algebra G_2

1998

This paper is a continuation of "Quantization of Lie bialgebras I-IV". The goal of this paper is to define and study the notion of a quantum vertex operator algebra in the setting of the formal deformation theory and give interesting examples of such algebras. In particular, we construct a quantum vertex operator algebra from a rational, trigonometric, or elliptic R-matrix, which is a quantum deformation of the affine vertex operator algebra. The simplest vertex operator in this algebra is the quantum current of Reshetikhin and Semenov-Tian-Shansky.

Journal of Physics A: Mathematical and General, 2006

The solutions of the Drinfeld equation corresponding to the full set of different carrier subalgebras f ⊂ sl3 are explicitly constructed. The obtained Hopf structures are studied. It is demonstrated that the presented twist deformations can be considered as limits of the corresponding quantum analogues (q-twists) defined for the q-quantized algebras.

Arxiv preprint q-alg/9605026, 1996

Czechoslovak Journal of Physics, 2005

We use the decomposition of o(3, 1) = sl(2; C) 1 ⊕ sl(2; C) 2 in order to describe nonstandard quantum deformation of o(3, 1) linked with Jordanian deformation of sl(2; C). Using twist quantization technique we obtain the deformed coproducts and antipodes which can be expressed in terms of real physical Lorentz generators. We describe the extension of the considered deformation of D = 4 Lorentz algebra to the twist deformation of D = 4 Poincaré algebra with dimensionless deformation parameter.

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).

Journal of Mathematical Physics, 2010

Exceptional modular invariants for the Lie algebras B 2 (at levels 2, 3, 7, 12) and G 2 (at levels 3, 4) can be obtained from conformal embeddings. We determine the associated algebras of quantum symmetries and discover or recover, as a by-product, the graphs describing exceptional quantum subgroups of type B 2 or G 2 which encode their module structure over the associated fusion category. Global dimensions are given.

Journal of Geometry and Physics, 2021

The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.

Selecta Mathematica, 1996

In the paper [Dr3] V.Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a universal quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of universal quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfeld's questions, using the methods and ideas of [KL]. In particular, we show the existence of a universal quantization. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.

1996

In this paper we construct explicitly the quantization of Lie bialgebras of a finite dimensional simple Lie algebra. by reducing the problem of quantization of the algebra of $\g$-valued functions on a curve with many punctures to the case of one puncture

Israel Journal of Mathematics, 2003

We prove that the reflection equation (RE) algebra L R associated with a finite dimensional representation of a quasitriangular Hopf algebra H is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that L R is a module algebra over the twisted tensor square H R ⊗ H and the double D(H). We define FRT-and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces. *