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

Abstract

We 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.