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Elliptic Algebras 

Paul Smith, University of Washington
Tuesday, October 8, 2019 - 2:30pm to 3:20pm
PDL C-36

Ongoing work with Alex Chirvasitu (SUNY, Buffalo) and Ryo Kanda (Osaka). This talk concerns the algebras $Q_{n,k}(E,\tau)$ defined by Odesskii and Feigin in 1989. Each is a connected graded ${\mathbb C}$-algebra, usually not commutative, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve $E={\mathbb C}/\Lambda$, and a translation automorphism $z \mapsto z+\tau$ of $E$. At first glance, its definition as the free algebra ${\mathbb C} \langle x_0,\ldots,x_{n-1}\rangle$ modulo the $n^2$ relations $$ \sum_{r \in {\mathbb Z}_n} \frac{\theta_{j-i+r(k-1)}(0)}{\theta_{j-i-r}(-\tau)\theta_{kr}(\tau)}\, x_{j-r} x_{i+r} \qquad (i,j) \in {\mathbb Z}_n^2 $$ reveals nothing. Here the $\theta_\alpha(z)$, $\alpha \in {\mathbb Z}_n$, are theta functions of order $n$ that are quasi-periodic with respect to the lattice $\Lambda$. For a fixed $(n,k,E)$ the $Q_{n,k}(E,\tau)$'s form a flat family of deformations of the polynomial ring ${\mathbb C}[x_0,\ldots,x_{n-1}]=Q_{n,k}(E,0)$. Establishing the basic properties of these algebras involves some interesting algebraic geometry. 

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