Classical Limit of genus two DAHA

Abstract

We show that one-parameter deformation ${\mathcal{A}}_{q,t}$ of the skein algebra $S{k}_q(\hat{I}{\pounds }_2)$ of a genus two surface suggested in [AS19] is flat. We solve the word problem in the algebra and describe monomial basis. In addition, we calculate the classical limit ${\mathcal{A}}_{q=1,t}$ of the algebra and prove that it is a one-parameter flat Poisson deformation of the coordinate ring ${\mathcal{A}}_{q=t=1}$ of an $SL(2,\mathbb{C})$-charater variety of a genus two surface. As a byproduct, we obtain a remarkably simple presentation in terms of generators and relations for the coordinate ring ${\mathcal{A}}_{q=t=1}$ of a genus two character variety.

37 pages, 1 figure