Noncommutative Symmetric Functions III: Deformations Of Cauchy And Convolution Algebras

This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shue product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in theoretical physics (the quon algebra)