Galatasaray University Mathematics Department Seminars

Noncommutative Algebraic Geometry on a Leavitt Path Algebra of Polynomial Growth
Murad Özaydın
University of Oklahoma, Turkey
Özet : Algebraic Geometry classically studies the geometry of sets given as the solutions to polynomial equations via the commutative algebra of "regular" functions on this set. According to the Gelfand-Grothendieck philosophy a commutative ring should be thought of as a ring of functions: Complex valued continuous functions for (locally) compact Hausdorff topological spaces (Gelfand-Naimark duality) where the points of the space correspond to maximal ideals; polynomial functions for affine varieties where we need all prime ideals (with the Zariski topology) to keep functoriality. In Connes's noncommutative geometry noncommutative rings are also regarded as rings of functions. Now there are several candidates for the "points": maximal ideals, primitive ideals and simple modules (these are equivalent when the ring is commutative). The general consensus is that there are never enough points (for instance to recover the original ring). Leavitt Path Algebras are constructed from the geometric data of a di(rected )graph G. A theorem of Alahmadi, Alsulami, Jain and Zelmanov says that they have polynomial growth if and only if the cycles of G are mutually disjoint. In this case there seem to be enough points (at least for algebraic quantum spheres) after some tweaking of the spectrum (= space of points). While this is a sequel to the seminar by Ayten Koc, familiarity with that talk is not a prerequisite. Relevant concepts will be (re)defined and graduate students are the target audience.
  Tarih : 30.11.2016
  Saat : 15:00
  Yer : FEF 9
  Dil : English