Related papers: Two-Parameter Quantum Groups and $R$-Matrices: Cla…
Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In the present article we present a parameterization of the unitary group…
We give a two-parameter quantum deformation of the exterior plane and its differential calculus without the use of any R-matrix and relate it to the differential calculus with the R-matrix. We prove that there are two types of solutions of…
Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce…
Quantum superalgebras $su_{q}(m\mid n)$ are studied in the framework of $R$-matrix formalism. Explicit parametrization of $L^{(+)}$ and $L^{(-)}$ matrices in terms of $su_{q}(m\mid n)$ generators are presented. We also show that quantum…
In this paper we establish affinizations and R-matrices in the language of pro-objects, and as an application, we construct reflection functors over the localizations of quiver Hecke algebras of arbitrary finite types. This reflection…
The recently obtained results in \cite{ZG2} are used to compute the explicitly spectral-dependent $R$-matrix (or the intertwiners) on $V_{(6)}(x)\otimes V_{(6)}(y)$ and $V_{(3)}(x)\otimes V_{(6)}(y)$, where $V_{(6)}$ and $V_{(3)}$ are the…
In this paper we explicitly attach to a geometric classical r-matrix $r$ (not necessarily unitary), a geometric (i.e., set-theoretical) quantum R-matrix $R$, which is a quantization of $r$. To accomplish this, we use the language of…
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain…
We give a formula for the universal R-matrix of the quantized universal enveloping algebra $U_q(\g).$ This is similar to a previous formula due to Kirillov-Reshetikhin and Levendorskii-Soibelman, except that where they use the action of the…
The quantum commutations $RTT=TTR$ and the orthogonal (symplectic) conditions for the inhomogeneous multiparametric $q$-groups of the $B_n,C_n,D_n$ type are found in terms of the $R$-matrix of $B_{n+1},C_{n+1},D_{n+1}$. A consistent Hopf…
We present a conjecture on the irreducibility of the tensor products of fundamental representations of quantized affine algebras. This conjecture implies in particular that the irreducibility of the tensor products of fundamental…
We detail the automatic construction of R matrices corresponding to (the tensor products of) the (0|\alpha) families of highest-weight representations of the quantum superalgebras U_q[gl(m|n)]. These representations are irreducible, contain…
The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical)…
On the basis of `$RTT=TTR$' formalism, we introduce the quantum double of the Yangian $Y_{\hbar}(\gtg)$ for $\gtg=\gtgl_N,\gtsl_N$ with a central extension. The Gauss decomposition of T-matrices gives us the so-called Drinfel'd generators.…
Integrable deformation of SU(2) sigma and lambda models are considered at the classical and quantum levels. These are the Yang-Baxter and XXZ-type anisotropic deformations. The XXZ type deformations are UV safe in one regime, while in…
We apply the general construction of a twist of bigraded Hopf algebras by skew bicharacters to obtain two-parameter quantum groups in the Drinfeld-Jimbo, new Drinfeld (for affine types), and FRT (for both finite and affine) presentations…
We study the two-parameter quantized enveloping algebra $U^+_{r,s}(B_2)$ at roots of unity and investigate its structure and representations. We first show that when $r$ and $s$ are roots of unity, the algebra becomes a PI algebra, and we…
For each simple Lie algebra $\mathfrak{g}$, we construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into certain quantum torus algebra $D_\mathfrak{g}$ via the positive representations of split real quantum group. The…
In this article we use a parametrized version of the FRT construction to construct two new coquasitriangular Hopf algebras. The first one, $\widehat{SL_q(2)}$, is a quantization of the coordinate ring on affine $SL(2)$. We show that there…
R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the…