Self-Dual Bases for Self-Dual Hopf Algebras
Advisor(s)
Dr. Aaron Lauve; Loyola University Chicago
Discipline
Mathematics
Start Date
21-4-2021 10:25 AM
End Date
21-4-2021 10:40 AM
Abstract
We consider when a graded, self-dual Hopf algebra A = A ,
with graded isomorphisms Φn: An → An* , has a self-dual basis. Motivated by the example of SSym over the reals, a self-dual algebra with no self-dual basis, we do this by analyzing the qualities of the map Φn. In particular, we have the general result that if this restricted map defines a real, symmetric, positive definite matrix, then A has a self-dual basis over C. We extend this to algebraically closed base fields, and show that SSym has a self-dual basis over C as a corollary.
Self-Dual Bases for Self-Dual Hopf Algebras
We consider when a graded, self-dual Hopf algebra A = A ,
with graded isomorphisms Φn: An → An* , has a self-dual basis. Motivated by the example of SSym over the reals, a self-dual algebra with no self-dual basis, we do this by analyzing the qualities of the map Φn. In particular, we have the general result that if this restricted map defines a real, symmetric, positive definite matrix, then A has a self-dual basis over C. We extend this to algebraically closed base fields, and show that SSym has a self-dual basis over C as a corollary.