Tetrahedron equation, Weyl group, and quantum dilogarithm

与量子化六顶点模型相关的四面体方程的新解

New solutions to the tetrahedron equation associated with quantized six-vertex models

本文构造了与量子化六顶点模型相关的一族四面体方程新解，分别基于q-振子代数和q-Weyl代数，后者给出可因式分解或表为截断q-超几何级数的新R矩阵。

量子簇代数与三维可积性：四面体方程与三维反射方程

Quantum cluster algebras and 3D integrability: Tetrahedron and 3D reflection equations

本文扩展量子簇代数方法，通过研究Fock-Goncharov箭图和簇变换，构造了四面体方程与三维反射方程的新解，并推导了其矩阵元的显式公式。

Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism we show how to construct integrable system with spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to general non-symmetric Newton polygons, and prove Lemma…

从与对称蝴蝶箭图相关的量子簇代数构造四面体方程的解

Solutions of Tetrahedron Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver

本文基于量子簇代数方法，利用对称蝴蝶箭图与 $q$-Weyl 代数构造了由量子双对数乘积组成的四面体方程新解，为多种已知的三维 $R$-矩阵提供了统一视角。

Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant

We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neumann-Zagier, we deform a complete hyperbolic structure of M, and correspondingly we define quantum invariant Z(M_u). This quantum invariant is shown to give the Neumann--Zagier potential function in the classical limit, and the A-polynomial can be derived from the potential functio…

The pentagon relation and incidence geometry

We define a map S: D^2 x D^2 --> D^2 x D^2, where D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev-P…

A Common Structure in PBW Bases of the Nilpotent Subalgebra of $U_q(\mathfrak{g})$ and Quantized Algebra of Functions

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new represent…

Positive Representations of Split Real Quantum Groups: The Universal R Operator

The universal $R$ operator for the positive representations of split real quantum groups is computed, generalizing the formula of compact quantum groups $U_q(g)$ by Kirillov-Reshetikhin and Levendorski\uı-Soibelman, and the formula in the case of $U_{q\tilde{q}}(sl(2,R))$ by Faddeev, Kashaev and Bytsko-Teschner. Several new functional relations of the quantum dilogarithm are obtained, generalizing the quantum exponential relations and the pentagon relations. The quantum Weyl element and Lusztig's isomorphism in the positive setting are also studied in detail. Finally we introduce a $C^*$-algeb…

Baxterization of GL_q(2) and its application to the Liouville model and some other models on a lattice

We develop the Baxterization approach to (an extension of) the quantum group GL_q(2). We introduce two matrices which play the role of spectral parameter dependent L-matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q-oscill…