Related papers: A new current algebra and the reflection equation
In [Kyushu J. Math. 64 (2010), 81-144, arXiv:0904.2889], it is discussed that a certain subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ controls the second kind TD-algebra of type I (the degenerate $q$-Onsager…
The non-standard quantum deformation of the (trivially) extended sl(2,R) algebra is used to construct a new quantum deformation of the two-photon algebra h_6 and its associated quantum universal R-matrix. A deformed one-boson representation…
This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this…
Using the previous obtained universal $R$-matrix for the quantized nontwisted affine Lie algebras $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for the corresponding quantum Lie…
A brief and elementary introduction to the subject is presented. Two open problems are given.
In this paper we construct a new factorized representation of the $R$-matrix related to the affine algebra $U_{q}(\widehat{sl_{n}})$ for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit…
We review the new approach to the theory of nonlinear $W$-algebras which is developed recently and called {\it conformal linearization}. In this approach $W$-algebras are embedded as subalgebras into some {\it linear conformal} algebras…
In this paper, we give an RTT presentation of the twisted quantum affine algebra of type $A_{2n-1}^{(2)}$ and show that it is isomorphic to the Drinfeld new realization via the Gauss decomposition of the L-operators. This provides the first…
In this paper, we recall our renormalized quantum Q-system associated with representations of the Lie algebra $A_r$, and show that it can be viewed as a quotient of the quantum current algebra $U_q({\mathfrak n}[u,u^{-1}])\subset…
The quantum group SL_q(2,R) at roots of unity is introduced by means of duality pairings with the quantum algebra U_q(sl(2,R)). Its irreducible representations are constructed through the universal T-matrix. An invariant integral on this…
We continue our study of Hilbert space representations of the Reflection Equation Algebra, again focusing on the algebra constructed from the $R$-matrix associated to the $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We develop a form…
The construction of a q-deformed N=2 superconformal algebra is proposed in terms of level 1 currents of ${\cal{U}}_{q} ({\widehat{su}}(2))$ quantum affine Lie algebra and a single real Fermi field. In particular, it suggests the expression…
We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on…
We introduce an analogue of the $q$-Schur algebra associated to Coxeter systems of type $\hat A_{n-1}$. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an…
We discuss new realisations of $W$ algebras in which the currents are expressed in terms of two arbitrary commuting energy-momentum tensors together with a set of free scalar fields. This contrasts with the previously-known realisations,…
This is a full revised version of the previous same titled article. The 2-cocycle of the central extension of the current algebra on the three sphere is taken the place of a new quarternion valued one, that is defined by the boundary Dirac…
In the context of connections between algebras coming from quantum integrable systems and algebras associated to the orthogonal polynomials of the Askey scheme, we prove that the truncated reflection algebra attached to the Yangian of sl(2)…
The quantum double construction of a $q$-deformed boson algebra possessing a Hopf algebra structure is carried out explicitly. The $R$-matrix thus obtained is compared with the existing literature.
We prove that the reflection equation (RE) algebra $\La_R$ associated with a finite dimensional representation of a quasitriangular Hopf algebra $\Ha$ is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We…
Following the ideas in~\cite{yM88} and some inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the…