# Is the quantum dilogarithm related in any way to cohomology of quantum groups?

Is the quantum dilogarithm related in any way to cohomology of quantum groups?

This question is a bit vague, and I don't have any rigorous justification for such a connection, but let me explain why morally there should be a connection.

1. Semisimple lie algebras have a natural Lie algebra $3$-cocycle (namely $B([\alpha,\beta],\gamma)$, where $B$ is the Killing form), so there is a natural class in $H^3(G;\mathbb C)$. This is essentially related to the Cheeger--Chern--Simons class which gives both the hyperbolic volume and Chern--Simons invariant of a flat $G$-connection, as well as the Chern--Simons functional used to construct the WRT TQFT. The corresponding map (roughly volume plus $i$ times Chern--Simons of an ideal simplex) on $H_3(\operatorname{SL}(2),\mathbb Z)$ can be described explicitly in terms of the (classical) dilogarithm. Thus we have the following classical connections: $$\hat c_2\in H^3(G)\iff\text{hyperbolic volume and Chern--Simons invariant}\iff\text{classical dilogarithm}$$

2. The volume conjecture predicts a rough analogue in the world of the WFT TQFT. The colored Jones polynomial is equal to Kashaev's knot invariant derived from the quantum dilogarithm. The Jones polynomial is intimately related to the Chern--Simons functional (which represents hyperbolic volume, and thus is related to the classical dilogarithm). Another motivating construction is the Quantum Hyperbolic Invariants of Baseilhac and Benedetti, which again give invariants of knot complements based on a quantum dilogarithm (which we might say is morally giving their "quantum volume").

3. Quantum groups also have cohomology (see my question Projective modules over quantum groups about this) agreeing roughly (perhaps exactly in most cases?) with the cohomology of the corresponding classical group. This, however, may be "too linear" to be related to the quantum dilogarithm (which is a sort of "deformed exponential" of the classical dilogarithm).

In summary, I expect the quantum dilogarithm to somehow "measure three-dimensional quantum volume", and relating it to third cohomology of quantum groups is just one way of making this intuition precise.

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I don't have a particular answer. I do know that a number of people "expect the quantum dilogarithm to somehow "measure three-dimensional quantum volume"". One name that comes to mind is Tudor Dimofte — you might start with his papers. –  Theo Johnson-Freyd Jan 16 '12 at 0:35