A bialgebra is, roughly speaking, an algebra on which there exists a dual
structure, called a coalgebra structure, such that the two structures satisfy
a compatibility relation. A Hopf algebra is a bialgebra with an endomorphism
satisfying a condition which can be expressed using the algebra and
coalgebra structures.
The first example of such a structure was observed in algebraic topology
by H. Hopf in 1941. This was the homology of a connected Lie group, which
is even a graded Hopf algebra. Starting with the late 1960s, Hopf algebras
became a subject of study from a strictly algebraic point of view, and by
the end of the 1980s, research in this field was given a strong boost by the
connections with quantum mechanics (the so-called quantum groups are in
fact examples of noncommutative noncocommutative Hopf algebras).