Related papers: Pentagon relation and Biedenharn-Elliott identity
We make a direct connection between the construction of three dimensional topological state sums from tensor categories and three dimensional quantum gravity by noting that the discrete version of the Wheeler-DeWitt equation is exactly the…
We show that solutions of Pentagon equations lead to solutions of the Tetrahedron equation. The result is obtained in the spectral parameter dependent case.
A set-theoretic solution to the Pentagon Equation can be described as a \emph{pentagon} algebra $(S, \cdot, \ast)$ such that $(S, \cdot)$ is a semigroup and the operations $\cdot$ and $\ast$ are related by two additional equations. This…
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…
It is shown that the Heisenberg double has a canonical element, satisfying the pentagon relation. From a given invertible constant solution to the pentagon relation one can restore the structure of the underlying algebras. Drinfeld double…
Representations of planar triangulations as contact graphs of a set of internally disjoint homothetic triangles or of a set of internally disjoint homothetic squares have received quite some attention in recent years. In this paper we…
The notion of a symmetrically factorizable Lie group is introduced. It is shown that each symmetrically factorizable Lie group is related to a set-theoretical solution of the pentagon equation. Each simple Lie group (after a certain Abelian…
We present a new solution to the pentagon identity in terms of gamma function. We obtain this solution by taking the gamma function limit from the pentagon identity related to the three-dimesional index. This limit corresponds to the…
We introduce a generalization of elliptic 6j-symbols, which can be interpreted as matrix elements for intertwiners between corepresentations of Felder's elliptic quantum group. For special parameter values, they can be expressed in terms of…
We establish some new theorems on pentagon and pentagram.
We extend the so-called retract relation given in [6] for involutive set-theoretic solutions of the Pentagon Equation and we introduce the notion of associated permutation group to study the family of the commutative non-degenerate ones.…
We introduce new algebraic structures associated with heptagon relations -- higher analogue of the well-known pentagon. The main points we deal with are: (i) polygon relations as algebraic imitations of Pachner moves, on the example of…
A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the…
A premonoidal category is equipped only with a bifunctor and a natural isomorphism for associativity. We introduce a (deformation) natural automorphism representing the deviation from the Pentagon condition. We uncover a binary tree…
The tetrahedron equation in a special substitution is reduced to a pair of pentagon and one ten-term equations. Various examples of solutions are found. $O$-doubles of Novikov, which generalize the Heisenberg double of a Hopf algebra,…
In [F] H. Furusho proves the beautiful result that of the three defining equations for associators, the pentagon implies the two hexagons (see also [W]). In this note we present a simpler proof for this theorem (although our paper is less…
This paper is on elimination of defining equations of the cyclotomic analogues, introduced by the first author, of Drinfeld's scheme of associators. We show that the mixed pentagon equation implies the octagon equation for N=2 and the…
We will attach a scalar invariant to a tetrahedron whose edges are labelled by irreducible representations of a ternary orthogonal group $\mathrm{SO}_3$ over a local field. This generalizes the $6j$ symbol whose theory was developed by…
We classify coproducts on matrix algebra in terms of solutions to some modification of pentagon equation. The construction of Baaj and Skandalis describing finite dimensional unitary solutions of pentagon equation is extended to the…
We consider a pentagon chain described by a Hubbard type of model considered under periodic boundary conditions. The system i) is placed in an external magnetic field perpendicular to the plane of the cells, and ii) is in a site selective…