New Biharmonic Functions on the Compact Lie Groups $$\mathbf{SO}(n)$$, $$\mathbf{SU}(n)$$, $$\mathbf{Sp}(n)$$

Abstract

AbstractWe develop 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 $$\mathbf{SU}(n)$$ SU ( n ) . We then show that the special orthogonal group $$\mathbf{SO}(n)$$ SO ( n ) and the quaternionic unitary group $$\mathbf{Sp}(n)$$ 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.