MARCELO AGUIAR AND JEAN-LOUIS LODAY
Abstract. We introduce the notion of quadri-algebras. These are associative alge-
bras 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: (T) = T id and (T) = id T, where denotes the convolution of endo-
morphisms. 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.
The study of the space of endomorphisms of an infinitesimal bialgebra revealed the
existence of peculiar algebraic structures. More specifically, the convolution of endo-
morphisms gives rise to a pair of commuting Baxter operators
(T) = T id and (T) = id T
and each of these determines a dendriform structure on the space of endomorphisms.