The universal quantum invariant and colored ideal triangulations

Abstract

The Drinfeld double of a finite dimensional Hopf algebra is a quasitriangular 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.

Cluster realization of $$\mathcal {U}_q(\mathfrak {g})$$ U q ( g ) and factorizations of the univers

“…In the case of Uq(sl2), such identification of the factorization of the R-matrix appears in quantum Teichmüller theory [30] as an element of the mapping class group, and the corresponding factorization is also used to re-derive Kashaev's knot invariant [16]. This has been utilized for example to construct new quantum invariant for "colored triangulations" of topological spaces recently [40]. Although the two factorizations are realized on different tensor spaces, we believe there is a strong connection between the two different factorizations, where A is identified with the Borel part of Uq(g), and it will be interesting to find an explicit relationship between them.…”

On the Structure of Set-Theoretic Polygon Equations

“…3 In [3], it was shown that the Wheeler-DeWitt equation for three-dimensional general relativity reduces to the pentagon relation. There is a comprehensive literature around the idea of representing Pachner moves of a triangulation in three dimensions by a solution of the pentagon equation (or a pentagon relation), see in particular [2,8,24,34,47,59]. In four dimensions, the dual hexagon equation plays a corresponding role [29] (also see Remark 6.2 below), in five dimensions it is the dual heptagon equation [35,36,37].…”

On Bar-Natan - van der Veen's perturbed Gaussians

“…In the last two decades the universal invariant has been extensively studied [Hab08,Suz10,Suz13,MS16,Suz18]. Although the definition looks harmless, computing this invariant may turn out to be a challenging task.…”