 
Summary: QUADRIALGEBRAS
MARCELO AGUIAR AND JEANLOUIS LODAY
Abstract. We introduce the notion of quadrialgebras. 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 quadrialgebras: 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 quadrialgebra 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 quadrialgebra, whenever A is an infinitesimal
bialgebra. We also discuss commutative quadrialgebras and state some conjectures
on the free quadrialgebra.
Introduction
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.
