Summary: Contemporary Mathematics
Infinitesimal Hopf algebras
Abstract. Infinitesimal bialgebras were introduced by Joni and Rota [J-R].
An infinitesimal bialgebra is at the same time an algebra and a coalgebra, in
such a way that the comultiplication is a derivation. In this paper we de-
fine infinitesimal Hopf algebras, develop their basic theory and present several
It turns out that many properties of ordinary Hopf algebras possess an
infinitesimal version. We introduce bicrossproducts, quasitriangular infinites-
imal bialgebras, the corresponding infinitesimal Yang-Baxter equation and a
notion of Drinfeld's double for infinitesimal Hopf algebras.
An infinitesimal bialgebra is a triple (A, m, ) where (A, m) is an associative
algebra, (A, ) is a coassociative coalgebra and for each a, b A,
(ab) = ab1b2 + a1a2b .
Infinitesimal bialgebras were introduced by Joni and Rota [J-R] in order to provide
an algebraic framework for the calculus of divided differences. Several new examples
are introduced in section 2. In particular, it is shown that the path algebra of an
arbitrary quiver admits a canonical structure of infinitesimal bialgebra.