Talk: Extremal equations and algebraic methods

Early Career Math Colloquium

Organizers: Rongchang Liu, Christian Parkinson, and Weinan Wang

November 27 @ 3:00 pm - 4:00 pm Eastern

The audience is invited to a perspective on reduction algebras, including algebraic methods described by Zhelobenko to find solutions of extremal equations. The general idea is to realize the solution space of a system of equations as the representation space of the system’s symmetry algebra (reduction algebra). In particular, the representation-theoretic definition of singular/primitive vectors induces a system of extremal equations; similarly, mathematical models of physical phenomena are characterized by extremal equations. Two reduction algebras arising from solving extremal equations are given by:

  • Considering the Laplace operator in dimension $n$ as an element of the $n$th Weyl algebra via a mapping from the Lie algebra $\mathfrak{sl}(2)$
  • Determining primitive vectors in tensor product representations of the Lie superalgebra $\mathfrak{osp}(1|2)$

The speaker’s collaboration with Jonas T. Hartwig has found success in the second route with the introduction of the diagonal reduction superalgebra of $\mathfrak{osp}(1|2)$. The first example runs throughout the writings of Zhelobenko and was the subject of Irmak Bukey’s undergraduate thesis, as supervised by the speaker. The pursuit of more examples of reduction algebras and their associated representation spaces as solution spaces to extremal equations is the focus of multiple ongoing projects.

YouTube recording

(Note that there are several typos and errors: Specifically, I did not distinguish between a sum over a basis of the space of primitive vectors, as the correct choice, from a sum over the entire space.)