# 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 Φ_{n}: A_{n }→ A_{n}^{* }, 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 Φ_{n}. 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 Φ_{n}: A_{n }→ A_{n}^{* }, 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 Φ_{n}. 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.