Classification of Lie bialgebras over current algebras

In the present paper we present a classification of Lie bialgebra structures on Lie algebras of type g[[u]] and g[u], where g is a simple finite dimensional Lie algebra.Comment: 26 page

New rr-Matrices for Lie Bialgebra Structures over Polynomials

For a finite dimensional simple complex Lie algebra $\mathfrak{g}$, Lie bialgebra structures on $\mathfrak{g}[[u]]$ and $\mathfrak{g}[u]$ were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to produce $r$-matrices which correspond to Lie bialgebra structures over polynomials

Classification of Low Dimensional Lie Super-Bialgebras

A thorough analysis of Lie super-bialgebra structures on Lie super-algebras osp(1|2) and super-e(2) is presented. Combined technique of computer algebraic computations and a subsequent identification of equivalent structures is applied. In all the cases Poisson-Lie brackets on supergroups are found. Possibility of quantizing them in order to obtain quantum groups is discussed. It turns out to be straightforward for all but one structures for super-E(2) group.Comment: 15 pages, LaTe

On Some Lie Bialgebra Structures on Polynomial Algebras and their Quantization

We study classical twists of Lie bialgebra structures on the polynomial current algebra $\mathfrak{g}[u]$, where $\mathfrak{g}$ 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 $\mathfrak{g}$. 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 $\mathfrak{sl}(n)$.Comment: 41 pages, LATE

Construction of a universal twist element from an R-matrix

We propose a method for construction of a universal twist element based on a constant quasi-classical unifary matrix solution of the Yang-Baxter equation. The method is applied to few known R-matrices corresponding to Lie (super) algebras of rank one. Bibliography: 13 titles. \ua9 2005 Springer Science+Business Media, Inc

Twists in U(sl(3)) and their quantizations

The solution of the Drinfeld equation corresponding to the full set of different carrier subalgebras in sl(3) 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.Comment: 31 pages, Latex 2e, to be published in Journ. Phys. A: Math. Ge

Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL2

We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer-Gervais quantization in terms of the corresponding double affine Hecke algebra.Comment: 6 page