Quantum coadjoint action and the $6j$-symbols of $U qsl 2$Report as inadecuate

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1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Abstract : We review the representation theory of the quantum group $U \varepsilon sl 2\mathbb{C}$ at a root of unity $\varepsilon$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories QHFT.
Our analysis relies on the quantum coadjoint action of De Concini-Kac-Procesi, and the theory of Heisenberg doubles of Poisson-Lie groups and Hopf algebras.
We identify the 6j-symbols of generic representations of $U \varepsilon sl2\mathbb{C}$, the main ingredients of QHFT, with a bundle morphism defined over a finite cover of the algebraic quotient $PSL 2\mathbb{C}-!-PSL 2\mathbb{C}$, of degree two times the order of $\varepsilon$.
It is characterized by a non Abelian 3-cocycloid identity deforming the fundamental five term relation satisfied by the classical dilogarithm functions, that relates the volume of hyperbolic 3-polyhedra under retriangulation, and more generally, the simplicial formulas of Chern-Simons invariants of 3-manifolds with flat $sl 2\mathbb{C}$-connections.

Keywords : quantum groups Poisson-Lie groups coadjoint action geometric invariant theory dilogarithm functions TQFT invariants of 3-manifolds

Author: Stéphane Baseilhac -

Source: https://hal.archives-ouvertes.fr/


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