Abstract
We show that there exist two natural endomorphism algebras for shuffle bialgebras such as Sh (X), where X is a graded set. One of these endomorphism algebras is a natural extension of the Malvenuto–Reutenauer Hopf algebra and is defined using graded permutations. The other one, the dendriform descent algebra, is a subalgebra of the first defined by mimicking the definition of the descent algebras by convolution from the graded projections in the tensor algebra. We study these algebras for their own, show that they carry bidendriform structures and establish freeness properties, study their generators, dimensions, bases, and also feature their relations to the internal structure of shuffle algebras. As an application of these ideas, we give a new proof of Chapoton's rigidity theorem for shuffle bialgebras.
Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
