CoZinbiel Hopf Algebras in Combinatorics
Open AccessMany combinatorial structures admit a notion of restriction. Linear orders restrict to suborders, graphs to vertex-induced subgraphs, and so on. Likewise, many combinatorial structures can be written as a disjoint union of "connected" structures. For example, every graph is a disjoint union of connected graphs, and every partition is a disjoint union of partitions with a single block. We use Joyal's theory of species to describe families of combinatorial objects with both a notion of restriction and a compatible notion of connected structures. Schmitt showed that if P is one such family then it gives rise to two connected, cocommutative Hopf algebras, K(P) and K(P). We study the primitive elements of these Hopf algebras. In particular, we describe a second basis for K(P), given by summing over a related partial order and show that this basis contains a basis for the primitive elements.The Hopf algebra K(P) is coZinbiel, meaning its coproduct can be written as the sum of two non-coassociative coproducts satisfying certain compatibility conditions. We employ this fact to define and study endomorphisms αi and iβ, which map into the primitives and are intimately related to the Dynkin idempotent. In particular, we show that α1 maps onto the primitive elements and the map 1β gives a basis for the free Lie algebra of primitives. Then we consider one-parameter deformations of the Hopf algebras K(P) and K(P), which are q-cotridendriform. We generalize the maps αi and iβ to maps SαU, T and SβU, T where S, U, and T are fixed, disjoint sets, and use this generalization to characterize the coradical filtration of K(P). We consider in more detail the special case where our family P of combinatorial objects is the set of all (simple) graphs and prove a number of results particular to this special case.Finally, just as every graded bialgebra gives rise to an associated descent algebra, every codendriform bialgebra gives rise to an associated dendriform descent algebra. We define this new dendriform algebra and show that it contains the map α1. We conclude by proving that the dendriform descent algebra of the tensor algebra T(V) is the free dendriform algebra on a single generator.
- All rights reserved
Notice to Authors
If you are the author of this work and you have any questions about the information on this page, please use the Contact form to get in touch with us.