Algebra & Combinatorics Research Group  |  Texas A&M University





A&C Seminar

Working Seminar
   in Algebra

All Seminars

A&C Seminar:  Fall 2008, Fridays, Milner 317, 3:00–3:50 p.m.

November 7 Sarah Witherspoon (TAMU)
3:00–3:50 Quantum Symmetric Algebras
Abstract:   A symmetric algebra on a vector space V is just a polynomial ring (in variables corresponding to a basis of V). This is a commutative algebra, and may be defined by generators in V, and relations yx = xy for all x and y in V. In this talk, we will put this example in a much bigger context, generalizing these relations to those coming from a braiding on tensor powers of V. The resulting algebras are called quantum symmetric algebras, or Nichols algebras, or Nichols-Woronowicz algebras. They have appeared in many places, including quantum groups, the cohomology of flag manifolds, and the recent classification, by Andruskiewitsch and Schneider, of finite dimensional pointed Hopf algebras. We will define quantum symmetric algebras and survey some of this history. Then we will present a series of Hopf algebras in positive characteristic, found in joint work with Aaron Lauve, via a new combinatorial approach to quantum symmetric algebras.