A Note on Quiver Yangians and $\mathcal{R}$-Matrices

For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation...

A new construction method of $R$-matrix is given. Let $A$ be a C$^{*}$-bialgebra with a comultiplication $\Delta$. For two states $\omega$ and $\psi$ of $A$ which satisfy certain conditions, we construct a unitary $R$-matrix $R(\omega,\psi)$ of the C$^{*}$-bialgebra $(A,\Delta)$ on the tensor product of GNS representation spaces associated with $\omega$ and $\psi$. The set $\{R(\omega,\psi):\omega,\psi\}$ satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontri...

This text provides an introduction and complements to some basic constructions and results in 2-representation theory of Kac-Moody algebras.

We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After having defined Poisson-Lie groups we study their relation to Lie-bi algebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantizatio...

An explicit quantization is given of certain skew-symmetric solutions of the classical Yang-Baxter, yielding a family of $R$-matrices which generalize to higher dimensions the Jordanian $R$-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa-Ueno Yang-Baxter operators on meromorphic functions; as boundary solutions of the quantum Yang-Baxter equation; via a vertex-IRF transformation from solutions to the dynamical ...

A universal $R$-matrix for the non-standard (Jordanian) quantum deformation of $sl(2,\R)$ is presented. A family of solutions of the quantum Yang--Baxter equation is obtained from some finite dimensional representations of this Lie bialgebra quantization of $sl(2,\R)$.

Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. In this paper, we investigate the relationship between the categories $\CC^0_{A_{N-1}^{(1)}}$ and $\CC^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur-Weyl duality functors $\F^{(t)}$ from $R^{A_\inft...

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.

We construct spectral parameter dependent R-matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral parameter dependent quantum Yang-Baxter equation.

This overview paper reviews several results relating the representation theory of quivers to algebraic geometry and quantum group theory. (Potential) applications to the study of the representation theory of wild quivers are discussed. To appear in the Proceedings of the International Conference on Representations of Algebras and Related Topics ICRA X, The Fields Institute, July/August 2002.