Modular double of a Quantum Group

Tetrahedron Equation and Quantum R Matrices for Modular Double of $${{\varvec{{U_q(D^{(2)}_{n+1})}}, \varvec{{U_q (A ^{(2)}_{2n})}}}}$$ U q ( D n + 1 ( 2 ) ) , U q ( A 2 n ( 2 ) ) and $$\varvec{{U_q(C^{(1)}_{n})}}$$ U q ( C n ( 1 ) )

“…An analogous feature has been observed in [11]. For general background on modular double, we refer to [8,9].…”

Matrix Factorization for Solutions of the Yang–Baxter Equation

“…we will prove the factorization formula (10). Besides the reduction formula (6) for sℓ 2 the analogous restrictions (to a finite-dimensional representation) of the general R-operator has been obtained in the paper [8] for the Lie group SL(2, C) and for the modular double of U q (sℓ 2 ) [17].…”

“…Thus we act sequentially on (z 1 − x) n Φ(z) by the operator factors from Eq. (17). At the first step we perform the shift…”

“…(1), that are invariant with respect to the modular double. The modular double of the quantum algebra U q (sℓ 2 ) has been introduced by Ludwig Faddeev in [17]. This algebra is formed by two sets of generators E , K , F and E , F , K. The standard commutation relations for the generators E , K , F, which form the quantum algebra U q (sℓ 2 ) with the deformation parameter q = e iπτ (we assume that τ ∈ C\Q, i.e.…”

“…In addition, the generators E and F commute with E and F; the generator K anticommutes with E and F; K anticommutes with E and F; K and K commute. The representation theory of the modular double has been elaborated in a number of papers, see for example [5,17,18,21,31] and references therein. We will use the following parametrization τ = …”