## Seminar in Cluster algebras

•  Leonid Chekhov, Michigan State University
•  Symplectic groupoid and cluster algebra description of closed Riemann surfaces
•  09/12/2022
•  12:30 PM - 1:30 PM
•  C304 Wells Hall
•  Linhui Shen (shenlin1@msu.edu)

We use the Fock-Goncharov higher Teichmuller space directed networks to solve the symplectic groupoid condition: parameterize pairs of $SL_n$ matrices (B,A) with A unipotent such that $BAB^T$ is also unipotent. A natural Lie-Poisson bracket on B generates the Goldman bracket on elements of A and $BAB^T$, which are simultaneously elements of the corresponding upper cluster algebras. Using this input we identify the space of X-cluster algebra elements with Teichmuller spaces of closed Riemann surfaces of genus 2 (for $n$=3) and 3 (for $n$=4) endowed with Goldman bracket structure: for $g$=2 all geodesic functions are positive Laurent polynomials and Dehn twists correspond to mutations in the corresponding quivers. This is the work in progress with Misha Shapiro.

## Contact

Department of Mathematics
Michigan State University
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C212 Wells Hall
East Lansing, MI 48824

Phone: (517) 353-0844
Fax: (517) 432-1562

College of Natural Science