# Elliptic Algebras

## Seminar/Forum

213
Peter Hall
These algebras, first defined in broad generality by Feigin and Odesskii, depend on an elliptic curve $$E$$, a translation automorphism $$\tau$$ of it (or, what is almost the same thing, a point $$\tau \in E$$), and a pair of relatively prime positive integers $$n>k>0$$. They are deformations of polynomial rings on $$n$$ variables. Their structure constants are expressed in terms of one-variable theta functions. When $$k=1$$ they are called Sklyanin algebras. Their behavior is intimately related to the geometry of the image of a certain embedding of $$E^p$$ in the $$(n-1)$$-dimensional projective space, where $$p$$ is the length of the negative continued fraction'' for $$n/k$$. This talk will discuss some of their properties found in joint work with Alex Chirvasitu and Ryo Kanda.