A proof of the pentagon relation for skeins

The (quantum) pentagon relation underlies the existing constructions of three dimensional quantum topology in the combinatorial framework of triangulations. Following the recent works \cite{KashaevLuoVartanov2012,AndersenKashaev2013}, we discuss a special type of integral pentagon relations and their relationships with the Faddeev type operator pentagon relations.

We construct vast families of orthogonal operators obeying pentagon relation in a direct sum of three n-dimensional vector spaces. As a consequence, we obtain pentagon relations in Grassmann algebras, making a far reaching generalization of exotic Reidemeister torsions.

This thesis explores a construction of the family of topological invariants for certain oriented 3-manifolds based on the state-integral approach developed by Andersen, Garoufalidis, and Kashaev in the Archimedean setting. Starting from an ideal triangulation of a 3-manifold equipped with angle data, variables are assigned to the faces and tetrahedra, taking values in a so-called 'Gaussian group'. The invariant is defined by integrating a distribution defined from the combinatorics of the triangulation and a special function over a product of the Gaussian group. The special function is a quantum dilogarithm, whose valuable feature, the pentagon relation, ensures the resulting integral remains unchanged under local modifications of the triangulation, thereby reflecting the topology of the manifold.

In this paper, we provide a combinatorial proof of the pentagon relations for valued quivers and their associated clusters for coecient free cluster algebras. Mathematics Subject Classication: 16S99 (primary); 68R99 (secondary)

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 $C^*$-algebraic version of the split real quantum group in the language of multiplier Hopf algebras, and consequently the definition of $R$ is made rigorous as the canonical element of the Drinfeld's double $\textbf{U}$ of certain multiplier Hopf algebra $\textbf{Ub}$. Moreover a ribbon structure is introduced for an extension of $\textbf{U}$.

We define a map \documentclass[12pt]{minimal}\begin{document}$S:{\mathbb {D}}^2\times {\mathbb {D}}^2 \dashrightarrow {\mathbb {D}}^2\times {\mathbb {D}}^2$\end{document}S:D2×D2⤏D2×D2, where \documentclass[12pt]{minimal}\begin{document}${\mathbb {D}}$\end{document}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–Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure—the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal $R$-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the $S$-tensor) satisfying the pentagon relation. In this paper we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of $3$-manifolds up to colored moves. In this construction, a copy of the $S$-tensor is attached to each tetrahedron, and invariance under the colored Pachner $(2,3)$ moves is shown by the pentagon relation of the $S$-tensor.

A full proof of Ocneanu’s theorem is given that one can produce a rational unitary polyhedral 3-dimensional topological quantum field theory of Turaev-Viro type from a subfactor with finite index and finite depth, and vice versa. The key argument is an equivalence between flatness of a connection in paragroup theory and invariance of a state sum under one of the three local moves of tetrahedra. This was announced by A. Ocneanu and he gave a proof of Frobenius reciprocity and the pentagon relation, which produces a 3-dimensional TQFT via the Turaev-Viro machinery, but he has not published a proof of the converse direction of the equivalence. Details are given here along the lines suggested by him.

It is shown that the tetrahedron equation under the substitution , where P 23 is the permutation operator, is reduced to a pair of pentagon and one ten-term equations on operators S and . Examples of infinite dimensional solutions are found. O-doubles of Novikov, which generalize the Heisenberg double of a Hopf algebra, provide a particular algebraic solution to the problem.

Isospin relations are used to eliminate hadronic uncertainties in various $\mathrm{CP}$ asymmetries in ${B}^{0}$ decays. In addition to the simple triangle relations for the $\ensuremath{\pi}\ensuremath{\pi}$ mode, we study quadrilateral relations for $K\ensuremath{\pi}$ and pentagon relations for $\ensuremath{\rho}\ensuremath{\pi}$. A combined angular and isospin analysis is required for $\ensuremath{\rho}\ensuremath{\rho}$. These methods are useful also for three-body decays such as $K\ensuremath{\pi}\ensuremath{\pi}$. The magnitude of the penguin amplitude can be extracted in various modes. The theoretical principles behind this analysis can be experimentally tested through sum rules for decay rates prior to the measurement of $\mathrm{CP}$ asymmetries.

Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants

This is a survey of the theory of quantum Teichmüller and Thurston spaces. The Thurston (or train track) theory is described and quantized using the quantization of coordinates for Teichmüller spaces of Riemann surfaces with holes. These surfaces admit a description by means of the fat graph construction proposed by Penner and Fock. In both theories the transformations in the quantum mapping class group that satisfy the pentagon relation play an important role. The space of canonical measured train tracks is interpreted as the completion of the space of observables in 3D gravity, which are the lengths of closed geodesics on a Riemann surface with holes. The existence of such a completion is proved in both the classical and the quantum cases, and a number of algebraic structures arising in the corresponding theories are discussed.

We give a proof of the pentagon relation for the quantum dilogarithm by using functional analysis methods. We introduce a related Schwartz space and prove that it is preserved by the intertwiner operator defined using the quantum dilogarithm. Using this we can define a representation of the quantized moduli space of configurations of 5 points on the projective line.

We consider a parameter-dependent version of the homotopy associative part of the Lian–Zuckerman homotopy algebra and provide an interpretation of the multilinear operations of this algebra in terms of integrals over certain polytopes. We explicitly prove the pentagon relation up to homotopy and propose a construction of the higher operations.

The (quantum) pentagon relations satisfied by the 6j-symbols of representation categories underlay the existing constructions in quantum topology based on combinatorics of triangulations of three-dimensional pseudo manifolds. We discuss the recent result as reported by Kashaev et al. (2012) on an integral five-term relation satisfied by Euler’s beta function.