  The Grothendieck group of the tower of symmetric group algebras has a
self-dual graded Hopf algebra structure. Inspired by this, we introduce by way
of axioms, a general notion of a tower of algebras and study two Grothendieck
groups on this tower linked by a natural paring. Using representation theory,
we show that our axioms give a structure of graded Hopf algebras on each
Grothendieck groups and these structures are dual to each other. We give some
examples to indicate why these axioms are necessary. We also give auxiliary
results that are helpful to verify the axioms. We conclude with some remarks on
generalized towers of algebras leading to a structure of generalized bialgebras
(in the sense of Loday) on their Grothendieck groups.
