Towards the Quantum Exceptional Series

Explore the fascinating world of Lie algebras and their quantum counterparts in this 57-minute lecture by Noah Snyder titled "Towards the Quantum Exceptional Series." Delve into the discrete families of Lie algebras such as GLn, On, and Spn, and discover how their corresponding planar algebras fit into continuous families like GLt and OSpt. Examine the work of Brauer, Deligne, and others in extending these concepts to quantum groups, introducing two-parameter families (GLt)q and (OSpt)q. Learn about the planar algebras associated with the HOMFLY and Kauffman polynomials. Investigate the exceptional Lie algebras G2, F4, E6, E7, and E8, and understand the conjectural one-parameter continuous family of planar algebras that interpolates between them. Explore the potential for a two-parameter family of planar algebras introducing a variable q, which could yield a new exceptional knot polynomial. Gain insights into joint work with Scott Morrison and Dylan Thurston, presenting a skein theoretic description of this hypothetical knot polynomial and the conditions necessary for braided tensor categories to satisfy exceptional skein relations.