### Dialgebras, by Jean-Louis Loday

There is a notion of "non-commutative Lie algebra" called Leibniz algebra,
which is characterized by the condition: left bracketing is a derivation.
The purpose of this article is to introduce and study a new notion of
algebra, called dialgebra, which is, to the Leibniz algebras, what
associative algebras are for Lie algebras. A dialgebra is a vector space
equipped with two associative operations satisfying three more conditions.
For instance, any differential associative algebra gives rise to a dialgebra.
In this article we construct and study a (co)homology theory for dialgebras.
The surprising fact, in the construction of the chain complex, is the
appearance of the combinatorics of planar binary trees (grafting and
nesting). The principal result about this homology theory is its vanishing
on free dialgebras.
The Koszul dual (in the sense of Ginzburg and Kapranov) of the operad of
dialgebras is the operad of dendric algebras. The dendric algebras are
characterized by two operations satisfying three linear conditions. The sum
of these two operations defines a new operation which is associative. The
free dendric algebras can be described in terms of planar binary trees. As a
consequence we give an explicitly description of strong homotopy dialgebras.
This paper is part of a long-standing project whose ultimate aim is to study
periodicity phenomenons in algebraic K-theory, as explained in "Overview on
Leibniz algebras, dialgebras and their homology". Fields Inst. Commun. 17
(1997), 91--102.

Jean-Louis Loday <loday@math.u-strasbg.fr>