Contents §0. Introduction §1. The Yangian §2. The quantum determinant and the centre of §3. The twisted Yangian §4. The Sklyanin determinant and the centre of §5. The quantum contraction and the quantum Liouville formula for the Yangian §6. The quantum contraction and the quantum Liouville formula for the twisted Yangian §7. The quantum determinant and the Sklyanin determinant of block matrices Bibliography

Yangians and Classical Lie Algebras

Hom-algebra structures are given on linear spaces by multiplications twisted by linear maps. Hom-Lie algebras and general quasi-Hom-Lie and quasi-Lie algebras were introduced by Hartwig, Larsson and Silvestrov as algebras embracing Lie algebras, super and color Lie algebras and their quasi-deformations by twisted derivations. In this paper we introduce and study Hom-associative, Hom-Leibniz and Hom-Lie admissible algebraic structures generalizing associative, Leibniz and Lie admissible algebras. Also, we characterize flexible Hom-algebras and explain some connections and differences between Hom-Lie algebras and Santilli’s isotopies of associative and Lie algebras.

Hom-algebra structures

A number of physical phenomena are described by nonlinear hyperbolic equations Presence of discontinuous solutions motivates the necessity of development of reliable numerical methods based on the fundamental mathematical properties of hyperbolic systems Construction of such methods for systems more complicated than the Euler gas dynamic equations requires the investigation of existence and uniqueness of the self-similar solutions to be used in the development of discontinuity-capturing high-resolution numerical methods This frequently necessitates the study of the behavior of discontinuities under vanishing viscosity and dispersion We discuss these problems in the application to the magnetohydrodynamic equations, nonlinear waves in elastic media, and electromagnetic wave propagation in magnetics

https://link.springer.com/content/pdf/10.1007%2F978-3-0348-8724-3_10.pdf

Mathematical Aspects of Numerical Solution of Hyperbolic Systems

In this paper we fill some gaps in the arguments of our previous papers [1,2] In particular, we give a proof that the L operators of Conformal Field Theory indeed satisfy the defining relations of the Yang–Baxter algebra Among other results we present a derivation of the functional relations satisfied by T and Q operators and a proof of the basic analyticity assumptions for these operators used in [1,2]

Integrable Structure of Conformal Field Theory III. The Yang–Baxter Relation

Status of the World’s Soil Resources Main report

Twisted Hopf algebra $sl_\xi(2)$ gives rise to a deformation of the Yangian ${\cal Y}(sl(2))$. The corresponding deformations of the integrable XXX-spin chain and the Gaudin model are discussed.

Deformed Yangians and Integrable Models

In 1982 Belavin and Drinfeld listed all elliptic and trigonometric solutionsX(u, v) of the classical Yang-Baxter equation (CYBE), whereX takes values in a simple complex Lie algebrag, and left the classification problem of the rational one open. In 1984 Drinfeld conjectured that if a rational solution is equivalent to a solution of the formX(u,v)=C2/(u−v)+r(u,v), whereC2 is the quadratic Casimir element andr is a polynomial inu,v, then degur=degvr≦1. In another paper I proved this conjecture forg=sl(n) and reduced the problem of listing “nontrivial” (i.e. nonequivalent toC2/(u−v)) solutions of CYBE to classification of quasi-Frobenius subalgebras of g. They, in turn, are related with the so-called maximal orders in the loop algebra of g corresponding to the vertices of the extended Dynkin diagramDe(g). In this paper I give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices ofDe(g) with coefficient 2 or 3. In particular I will find all solutions forg=o=(5) and some solutions forg=o(7),o(10),o(14) andg2.

/pdf/on-rational-solutions-of-yang-baxter-equations-maximal-1xry0jb0gd.pdf

On rational solutions of Yang-Baxter equations. Maximal orders in loop algebra

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.

Gauss decomposition of trigonometric R-matrices

An approach to Hopf algebras via Frobenius coordinates.

http://www.math.chalmers.se/Math/Research/Preprints/2001/10.pdf

An approach to hopf algebras via frobenius coordinates ii

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