Gauss decomposition of trigonometric R-matrices

The general formula for the universal R-matrix for quantized nontwisted affine algebras by Khoroshkin and Tolstoy is applied for zero central charge highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonomitric R-matrix in tensor product of these modules. Explicit calculations for the simplest case of $A_1^{(1)}$ are presented. As a consequence new formulas for the trigonometric R-matrix with a parameter in tensor product of $U_q(sl_2)$-Verma modules are obtained.Comment: 14 page

Chains of Frobenius subalgebras of so(M) and the corresponding twists

Chains of extended jordanian twists are studied for the universal enveloping algebras U(so(M)). The carrier subalgebra of a canonical chain F cannot cover the maximal nilpotent subalgebra N(so(M)). We demonstrate that there exist other types of Frobenius subalgebras in so(M) that can be large enough to include N(so(M)). The problem is that the canonical chains F do not preserve the primitivity on these new carrier spaces. We show that this difficulty can be overcome and the primitivity can be restored if one changes the basis and passes to the deformed carrier spaces. Finally the twisting elements for the new Frobenius subalgebras are explicitly constructed. This gives rise to a new family of universal R-matrices for orthogonal algebras. For a special case of g = so(5) and its defining representation we present the corresponding matrix solution of the Yang-Baxter equation.Comment: 17 pages, Late

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

Classification of quantum groups and Belavin--Drinfeld cohomologies for orthogonal and symplectic Lie algebras

In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.Comment: 17 page

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