${\mathfrak{sl}}_3$ Matrix Dilogarithm as a $6j$-Symbol

Abstract

We construct quantum invariants of 3-manifolds based on a ${\mathfrak{sl}}_3$ matrix dilogarithm proposed by Kashaev. This matrix dilogarithm is an ${\mathfrak{sl}}_3$ analogue of the (cyclic) quantum dilogarithm used to define Kashaev's invariants as well as Baseilhac and Benedetti's quantum hyperbolic invariants. % In this article, we show that the ${\mathfrak{sl}}_3$ matrix dilogarithm can be considered as a 6$j$-symbol associated to modules of a quantum group related to $U_q({\mathfrak{sl}}_3)$. Moreover, we show that the quantum invariants aforementioned allow to define a ${\mathfrak{sl}}_3$ version of Kashaev's invariants, opening a route to define a ${\mathfrak{sl}}_3$ version of Baseilhac and Benedetti's quantum hyperbolic invariants.

arXiv admin note: text overlap with arXiv:1008.3103 by other authors