Related papers: Quantum groups, Yang-Baxter maps and quasi-determi…
In this paper, we first introduce the notion of projective Banach Lie bialgebras as the projective tensor product analogue of Banach Lie bialgebras. Then we consider the completion of the classical Yang-Baxter equation and classical…
The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant…
The quantum dynamical Yang-Baxter (or Gervais-Neveu-Felder) equation defines an R-matrix R(p), where $p$ stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the…
Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding…
We develop a new approach to Baxter Q-operators by relating them to the theory of Yangians, which are the simplest examples for quantum groups. Here we open up a new chapter in this theory and study certain degenerate solutions of the…
A mathematically rigorous relativistic quantum Yang-Mills theory with an arbitrary semisimple compact gauge Lie group is set up in the Hamiltonian canonical formalism. The theory is non-perturbative, without cut-offs, and agrees with the…
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum…
A fundamental construction of Poisson algebras is to derive them as the quasiclassical limits (QCLs) of associative algebra deformations of commutative associative algebras. This paper lifts this process to the level of classical…
The type-I quantum superalgebras are known to admit non-trivial one-parameter families of inequivalent finite dimensional irreps, even for generic $q$. We apply the recently developed technique to construct new solutions to the quantum…
We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finite-dimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sl_2, the…
The elements of the wide class of quantum universal enveloping algebras are prooved to be Hopf algebras $H$ with spectrum $Q(H)$ in the category of groups. Such quantum algebras are quantum groups for simply connected solvable Lie groups…
Yang-Baxter type models are integrable deformations of integrable field theories, such as the principal chiral model on a Lie group $G$ or $\sigma$-models on (semi-)symmetric spaces $G/F$. The deformation has the effect of breaking the…
This talk is inspired by two previous ICM talks, by V.Drinfeld (1986) and G.Felder (1994). Namely, one of the main ideas of Drinfeld's talk is that the quantum Yang-Baxter equation (QYBE), which is an important equation arising in quantum…
We obtain a closed form expression of the universal T-matrix encapsulating the duality of the quantum superalgebra U_q[osp(1/2)] and the corresponding supergroup OSp_q(1/2). The classical q-->1 limit of this universal T-matrix yields the…
We present an uniform construction of the solution to the Yang- Baxter equation with the symmetry algebra $s\ell(2)$ and its deformations: the q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator acting in the…
Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian $\mathfrak{gl}_{1}$. The characteristic feature of the algebra is the action on BPS states for non-compact…
We consider a special class of quantum non-dynamical $R$-matrices in the fundamental representation of ${\rm GL}_N$ with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case $N=2$…
Starting with a four-dimensional gauge theory approach to rational, elliptic, and trigonometric solutions of the Yang-Baxter equation, we determine the corresponding quantum group deformations to all orders in $\hbar$ by deducing their RTT…
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any…
We construct a new class of quantum vertex algebras associated with the normalized Yang $R$-matrix. They are obtained as Yangian deformations of certain $\mathcal{S}$-commutative quantum vertex algebras and their $\mathcal{S}$-locality…