Title: Quantum cluster algebras from geometry
Speaker: Leonid Chekhov (Michigan State University)
Abstract: We identify the Teichmuller space $T_{g,s,n}$ of (decorated) Riemann surfaces $\Sigma_{g,s,n}$ of genus $g$, with $s>0$ holes and $n>0$ bordered cusps located on boundaries of holes uniformized by Poincare with the character variety of $SL(2,R)$-monodromy problem. The effective combinatorial description uses the fat graph structures; we can construct all observables, which are geodesic functions of closed curves and $\lambda$-lengths of paths starting and terminating at bordered cusps decorated by horocycles, out of extended shear coordinates; the Poisson and quantum algebras of observables are then induced by the Poisson and quantum structures of the extended shear coordinates; for $\lambda$-lengths we obtain the quantum cluster algebras of Berenstein and Zelevinsky. A seed of the corresponding quantum cluster algebra corresponds to the partition of $\Sigma_{g,s,n}$ into ideal triangles, $\lambda$-lengths of their sides are cluster variables constituting a seed of the algebra; their number $6g-6+3s+2n$ (and, correspondingly, the seed dimension) coincides with the dimension of $SL(2,R)$-character variety given by $[SL(2,R)]^{2g+s+n-2}/\prod_{i=1}^n B_i$, where $B_i$ are Borel subgroups associated with bordered cusps. Moreover, we construct quantum $SL(2,R)$- and $SL(2,C)$-monodromy matrices themselves; the corresponding Poisson and quantum algebras can be presented in an R-matrix form; they coincide with the Fock-Rosly algebras of monodromies. The talk is based on the joint papers with with M.Mazzocco and V.Roubtsov.
Seminar URL: https://people.math.osu.edu/anderson.2804/gcis/