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.
Professor Paul Smith, University of Washington in Seattle, Washington