  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: $\beta(T)=T\ast\id$ and
$\gamma(T)=\id\ast T$, where $\ast$ 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.
