arXiv subjects
Publications and source records attributed to V. Tolstoy.
We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang--Baxter equation. The method is based on an affine realization of certain seaweed algebras and their quantum analogues. We also propose a method of $ω$-affinization, which enables us to quantize rational $r$-matrices of $\mathfrak{sl}(3)$.
We find certain functional identities for the Gauss q-power function of a sum of q-commuting variables. Then we use these identities to obtain two-parameter twists of the quantum affine algebra U_q (\hat{sl}_2) and of the Yangian Y(sl_2). We determine the corresponding deformed trigonometric and rational quantum R-matrices, which then are used in the computation of deformed XXX and XXZ Hamiltonians.
The classical limit of the scaled elliptic algebra $A_{\hbar,η}(sl_2)$ is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic $sl_2$ valued functions on a strip and as an extended algebra of decreasing automorphic $sl_2$ valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $A_{q,p}(sl_2)$ is also briefly presented.