The centre of the graph and moduli algebras at roots of 1

The structure of the centres ${ \cal Z}(\Lg)$ and ${ \cal Z}(\Mg)$ of the graph algebra ${ \cal L}_g(sl_2)$ and the moduli algebra ${ \cal M}_g(sl_2)$ is studied at roots of 1. It it shown that ${ \cal Z}(\Lg)$ can be endowed with the structure of the Poisson graph algebra. The elements of $Spec({ \cal Z}(\Mg))$ are shown to satisfy the defining relation for the holonomies of a flat connection along the cycles of a Riemann surface. The irreducible representations of the graph algebra are constructed.

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