Algebra Seminar- S. Paul Smith, University of Washington
4:00PM, Mon Dec 3 2018, 150 Mathematics Bldg.
The algebras of the title form a flat family of (non-commutative!)
deformations of polynomial rings. They depend on a relatively prime
pair of integers n>k>0, an elliptic curve E, and a translation
automorphism of E. Quite a lot is known when n=3 and n=4 (and k=1),
in which case the algebras are deformations of the polynomial ring on
3 and 4 variables. These were discovered and have been closely studied
by Artin, Schelter, Tate, and Van den Bergh, and Sklyanin. They were
defined in full generality by Feigin and Odesskii around 1990 and
apart from their work at that time they have been little studied.
Their representation theory appears to be governed by, and best
understood in terms of, the geometry of embeddings of powers of E (and
related varieties like symmetric powers of E) in projective
spaces. Theta functions in several variables and mysterious identities
involving them provide a powerful technical tool.
This is a report on joint work with Alex Chirvasitu and Ryo Kanda.