2003/09/09 by Marcelo Aguiar, Jean-Louis Loday, Aguiar, Marcelo +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #math.QA #math.RA #msc:17A30 #msc:18D50
paper · pdf · doi:10.48550/arxiv.math/0309171
arxiv created 2004/04/05 · arxiv updated 2009/12/01
We introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. We present several examples of quadri-algebras: the algebra of permutations, the shuffle algebra, tensor products of dendriform algebras. We show that a pair of commuting Baxter operators on an associative algebra gives rise to a canonical quadri-algebra structure on the underlying space of the algebra. The main example is provided by the algebra End(A) of linear endomorphisms of an infinitesimal bialgebra A. This algebra carries a canonical pair of commuting Baxter operators: β(T)=T∗\id and γ(T)=\id∗ T, where ∗ denotes the convolution of endomorphisms. It follows that End(A) is a quadri-algebra, whenever A is an infinitesimal bialgebra. We also discuss commutative quadri-algebras and state some conjectures on the free quadri-algebra.