Cambrian Hopf algebras

Cambrian trees are oriented and labeled trees which fulfill local conditions around each node generalizing the classical conditions for binary search trees. Similar to binary trees for the Tamari lattice, Cambrian trees provide convenient combinatorial models for N. Reading's Cambrian lattices of type A. Based on a natural surjection from signed permutations to Cambrian trees, we define the Cambrian Hopf algebra extending J.-L. Loday and M. Ronco's algebra on binary trees. We describe combinatorially the products and coproducts of both the Cambrian algebra and its dual in terms of operations on Cambrian trees. We then construct the Baxter–Cambrian algebra which extends S. Law and N. Reading's Baxter Hopf algebra on rectangulations and S. Giraudo's equivalent Hopf algebra on twin binary trees.