# New biharmonic functions on the compact Lie groups $SO(n)$, $SU(n)$, $Sp(n)$

## TL;DR

This paper introduces a new method for constructing explicit proper biharmonic functions on Riemannian Lie groups, producing numerous new solutions on groups like $SU(n)$, $SO(n)$, and $Sp(n)$, and deriving new harmonic morphisms.

## Contribution

The paper develops a novel scheme for explicit biharmonic function construction on Lie groups, expanding the set of known solutions on classical groups.

## Key findings

- Produced infinite series of new biharmonic functions on $SU(n)$
- Extended the scheme to $SO(n)$ and $Sp(n)$ groups
- Derived new harmonic morphisms on these groups

## Abstract

We develope a new scheme for the construction of explicit complex-valued proper biharmonic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special unitary group $SU(n)$. We then show that the special orthogonal group $SO(n)$ and the quaternionic unitary group $Sp(n)$ fall into the scheme. As a by-product we obtain new harmonic morphisms on these groups. All the constructed maps are defined on open and dense subsets of the corresponding spaces.