Weighted Infinitesimal Unitary Bialgebras,Pre-Lie,Matrix Algebras And Polynomial Algebras

We construct an infinitesimal unitary bialgebra on matrix algebras and a weighted infinitesimal unitary bialgebra on non-commutative polynomial algebras,via two constructions of coproducts.Furthermore,an infinitesimal unitary Hopf algebra,under the view of Aguiar,is constructed on a matrix algebra.By investigating the relationship between weighted infinitesimal(unitary)bialgebras and pre-Lie algebras,we erect a pre-Lie algebra and a new Lie algebra on matrix algebras.Finally,a pre-Lie algebra and a Lie algebra on non-commutative polynomial algebras are also given.