These lecture notes give an introduction to bialgebras, algebraic objects arising in parts of mathematics, especially in topology. Roughly speaking, a bialgebra over a field \(k\) is a vector space \(B\) with two operations a product \(\mu\colon B\otimes B\to B\) and a coproduct \(\Delta\colon B\to B\otimes B\). These operations have to satisfy some natural conditions. The joint existence of two operations leads to a very interesting and nontrivial theory.
In Sections 1-3 of his notes the author gives a preparatory study of coalgebras, in which only the coproduct \(\Delta\) exists. Section 4 contains the definition of bialgebras and their simplest properties. An element \(x\in B\) is said to be primitive if \(\Delta x=1\otimes x+x\otimes 1\). The author proves that the set \(P_1(B)\) of primitive elements in \(B\) is closed under the commutation operation \(xy-yx\) and under the operation of taking the \(p\)th power if \(p=\text{char\,}k\neq 0\). So \(P_1(B)\) becomes a Lie algebra [a restricted Lie algebra if \(\text{char\,}k\neq 0\)]. In Section 5 the author proves that under some conditions imposed on bialgebras \(B\) and \(C\) any Lie algebra isomorphism \(P_1(B)\to P_1(C)\) has a unique extension to a bialgebra isomorphism \(B\to C\). Section 6 contains definition and properties of so-called Hopf algebras. Finally in Section 7 the author studies the structure of bialgebras whose underlying coalgebra is a coalgebra of divided powers in one variable. Such bialgebras arise from one-dimensional formal groups. Two appendices contain a list of all bialgebras of dimension \(\leq 4\) and a number of open questions and conjectures. A large bibliography is included.