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Tetrahedron equation and quantum cluster algebras

Rei Inoue, Atsuo Kuniba, Yuji Terashima

Abstract

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y-variables in terms $q$-Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's $R$ matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when $q$ is specialized to a root of unity.

Tetrahedron equation and quantum cluster algebras

Abstract

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y-variables in terms -Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when is specialized to a root of unity.
Paper Structure (24 sections, 10 theorems, 107 equations, 8 figures, 1 table)

This paper contains 24 sections, 10 theorems, 107 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum cluster algebra
  3. Mutation
  4. Tropical $y$-variables and tropical sign
  5. Sequence of mutations
  6. A useful theorem
  7. Cluster transformation $\widehat{R}$
  8. Wiring diagram and square quiver
  9. Cluster transformation $\widehat{R}$ as composition of mutations
  10. $\widehat{R}$ satisfies the tetrahedron equation
  11. Monomial solutions to the tetrahedron equation
  12. Dilogarithm identities
  13. Realization in terms of $q$-Weyl algebra
  14. Y-variables and $q$-Weyl algebra
  15. Extracting $R_{123}$ from $\widehat{R}_{123}$
  16. ...and 9 more sections

Key Result

Theorem 2.2

Let $(B',y') = \mu_{i_L} \cdots \mu_{i_2} \mu_{i_1}(B,u)$ be a tropical $y$-seed with $y'=(y'_i)_{i \in I}$. For any sequence $(i_1,\ldots, i_L) \in I^L$, each $y'_i \in \mathbb{P}(u)$ is either positive or negative.

Figures (8)

  • Figure 3.1: Wiring diagrams for the reduced words 212 and 121 of the longest element $s_2s_1s_2=s_1s_2s_1$ of $W(A_2)$.
  • Figure 3.2: Square quivers (depicted in blue) associated with the wiring diagrams. Given the labels $1,\ldots, 9$ of the quiver vertices in (a), those in (b) are determined following the mutation sequence in Figure \ref{['fig:mus']}.
  • Figure 3.3: The quivers $B=B^{(1)}, \ldots, B^{(6)}=B'$ and the mutations connecting them. We do not consider the wiring diagrams corresponding to the intermediate ones $B^{(2)}, \ldots, B^{(5)}$.
  • Figure 3.4: Cluster transformation $\widehat{R}_{123}$, which acts on the $q$-Weyl variables attached to the vertices $1,2,3$ of the wiring diagram colored in red.
  • Figure 3.5: Cluster transformations $\widehat{R}_{ijk}$. The wiring diagrams have 6 crossings (red). The quivers (blue) have 16 vertices. The seeds $(B^{(t)}, Y^{(t)})$ and $(B^{(t)\prime}, Y^{(t)\prime})$ will be explained in detail in Section \ref{['ss:ms']}.
  • ...and 3 more figures

Theorems & Definitions (23)

  • Example 2.1
  • Theorem 2.2: FZ07GHKK14
  • Remark 2.3
  • Theorem 2.4
  • Remark 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 13 more