arXiv subjects

A. A. Stolin

Publications and source records attributed to A. A. Stolin.

7 recordsLinked to original sources

arXiv2007

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 $\mathfrak{sl}(n)$.

math.QA↗

arXiv2003

On construction of universal twist element from $R$-matrix

A method to construct the universal twist element using the constant quasiclassical unitary matrix solution of the Yang - Baxter equation is proposed. The method is applied to few known $R$ -matrices, corresponding to Lie (super) algebras of rank one.

math.QA↗

arXiv2001

An Approach to Hopf Algebras via Frobenius Coordinates I

In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H \cite{Par72} and extend two recent theorems in \cite{EG}. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in \cite{FMS}. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular.

math.RA↗

arXiv2001

An approach to Hopf algebras via Frobenius coordinates II

We study a Hopf algebra $H$, which is finitely generated and projective over a commutative ring $k$, as a $P$-Frobenius algebra. We define modular functions in this setting, and provide a complete proof of Radford's formula for the fourth power of the antipode, using Frobenius algebraic techniques. As further applications, we extend Etingof and Gelaki's result that a separable and coseparable Hopf algebra has antipode of order two, the result of Schneider that Hopf subalgebras are twisted Frobenius extensions, and show that the quantum double is always a Frobenius algebra.

math.RA↗

arXiv1997

Deformed Yangians and Integrable Models

Twisted Hopf algebra $sl_ξ(2)$ 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.

q-alg↗

arXiv1995

Deformation of Yangian $Y(sl_2)$

A quantization of a non-standard rational solution of CYBE for $sl_2$ is given explicitly. We obtain the quantization with the help of a twisting of the usual Yangian $Y(sl_2$. This quantum object (deformed Yangian $Y_{η,ξ}(sl_2))$ is a two-parametric deformation of the universal enveloping algebra $U(sl_2[u])$ of the positive current algebra $sl_2[u]$. We consider the pseudotriangular structure on $Y_{η,ξ}(sl_2)$, the quantum double $DY_{η,ξ}(sl_2)$ its the universal R-matrix and also the RTT-realization of $Y_{η,ξ}(sl_2)$.

q-alg↗

arXiv1994

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.

hep-th↗