Algebra Seminar

Speaker: Sheldon Joyner (Western)

"On a certain $PSL(2, Z) 1$-cocycle"

Time: 14:30

Room: MC 107

Classically, if some manifold $M$ is equipped with an action of a subgroup $G$ of $PSL(2, Z)$ under which a certain space $F$ of functions on $M$ transforms via a $1$-cocycle, the latter is referred to as an automorphy factor, and the functions $F$ are said to be $G$-modular. In this talk, I will produce an injective $1$-cocycle of $PSL(2, Z)$ into a certain group of formal power series which extends the well-known identification of the fundamental group of $P^1\{0,1,\infty\}\;$ with associated Chen series. This cocycle may be regarded as a quasi-automorphy factor for sections of the universal prounipotent bundle with connection on $PSL(2, Z)$ - in particular for the polylogarithm generating series $Li(z)$. I will go on to show that the quasi-modularity of $Li(z)$ may be used to give a family of proofs of the analytic continuation and functional equation for the Riemann zeta function. Moreover, under this cocycle, the involutive generator of $PSL(2, Z)$ maps to the Drinfeld associator, while the infinite cyclic generator maps to an $R-matrix$, in Drinfeld's formal model of the quasi-triangular quasi-Hopf algebras, thereby producing a representation of $PSL(2, Z)$ into tensor products of certain $qtqH$ algebras. Underlying the whole story is a path space realization of $PSL(2, Z)$ using Deligne's idea of tangential basepoint.