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A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions…
We present a generalization of the master solution to the quantum Yang-Baxter equation (obtained recently in arXiv:1006.0651) to the case of multi-component continuous spin variables taking values on a circle. The Boltzmann weights are…
The quantum Yang-Baxter equation is a braiding condition on vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. Their combinatorial counterpart are set-theoretic solutions…
Every rack $Q$ provides a set-theoretic solution $c_Q$ of the Yang-Baxter equation. This article examines the deformation theory of $c_Q$ within the space of Yang-Baxter operators over a ring $\A$, a problem initiated by Freyd and Yetter in…
In the first part, we focus on indecomposable involutive solutions of the Yang-Baxter equation whose permutation group forces them to be uniconnected. Indecomposable involutive solutions with a permutation group isomorphic to a dihedral…
In this paper we study twisted traces of products of intertwining operators for quantum affine algebras. They are interesting special functions, depending on two weights lambda, mu, three scalar parameters q, omega, k, and spectral…
This paper combines mathematical, philosophical, and historical analyses in a comprehensive investigation of the dynamical foundations of the formalism of orthodox quantum mechanics. The results obtained include: (i) A deduction of the…
We propose a differential representation for the operators satisfying the q-mutation relation $BB^\dagger-q B^\dagger B=\1$ which generalizes a recent result by Eremin and Meldianov, and we discuss in detail this choice in the limit…
We give an explicit formula for the exchange matrix correponding to the tensor product of two copies of the natural (standard) evaluation representation of the quantum group associated to the affine Lie algebra of sl(n+1). Then we calculate…
We show that twisted reduced models can be interpreted as noncommutative Yang-Mills theory. Based upon this correspondence, we obtain noncommutative Yang-Mills theory with D-brane backgrounds in IIB matrix model. We propose that IIB matrix…
The principles of the theory of quantum groups are reviewed from the point of view of the possibility of their use for deformations of symmetries in physical models. The R-matrix approach to the theory of quantum groups is discussed in…
In the previous paper we showed that the wave functions of the quantum Ruijsenaars hyperbolic system diagonalize Baxter Q-operators. Using this property and duality relation we prove orthogonality and completeness relations for the wave…
Consider the statement "Every Yang-Baxter integrable system is defined to be exactly-solvable". To formalise this statement, definitions and axioms are introduced. Then, using a specific Yang-Baxter integrable bosonic system, it is shown…
In an earlier paper, two of the authors defined a $5$-vertex Yang-Baxter algebra (a Hopf algebra) which acts on the sum of the equivariant quantum K-rings of Grassmannians $\mathrm{Gr}(k;n)$, where $k$ varies from $0$ to $n$. We construct…
The definitions of the main notions related to the quantum inverse scattering methods are given. The Yang-Baxter equation and reflection equations are derived as consistency conditions for the factorizable scattering on the whole line and…
Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and…
An operator deformed quantum algebra is discovered exploiting the quantum Yang-Baxter equation with trigonometric R-matrix. This novel Hopf algebra along with its $q \to 1$ limit appear to be the most general Yang-Baxter algebra underlying…
Yangian symmetry of amplitudes in $\mathcal{N}=4$ super Yang-Mills theory is formulated in terms of eigenvalue relations for monodromy matrix operators. The Quantum Inverse Scattering Method provides the appropriate tools to treat the…
We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for…
This brief review surveys recent progress driven by the gauge/Yang-Baxter equation (YBE) correspondence. This connection has proven to be a powerful tool for discovering novel integrable lattice spin models in statistical mechanics by…