Math 6329 Hopf algebras and applications


Hopf algebras appeared around 1950 in two different areas: topology and
algebraic groups in positive characteristic. After a period of study of
commutative and cocommutative Hopf algebras, the development of the
general theory of Hopf algebras started around 1965. A breakthrough came
in the early 1980s with the introduction of quantum groups by Drinfel'd
and Jimbo. (In spite of the name, these fundamental objects are not
groups, but in fact Hopf algebras.)

Hopf algebras find applications in many branches of mathematics and
mathematical physics, including representation theory, low-dimensional
topology, conformal field theory, and combinatorics. Recently there has
been significant progress in the classification of Hopf algebras, which
has put the quantum groups of Drinfel'd and Jimbo into the context of a
general theory.


The objective of the course is to introduce the student to the theory of
Hopf algebras. Usually, covered topics include the structure theorem for
Hopf modules, the theory of integrals in Hopf algebras, Radford's
formula for the fourth power of the antipode, the Nichols-Zoeller
theorem on freeness of finite-dimensional Hopf algebras over Hopf
subalgebras, the relation between semisimplicity and cosemisimplicity,
the classification of Hopf algebras of prime dimension, quasitriangular
Hopf algebras, and the Drinfel'd double construction.


Advanced Linear Algebra (6351) or equivalent.


  • S. Dascalescu/C. Nastasescu/S. Raianu: Hopf algebras: An introduction,
    Pure Appl. Math., Vol. 235, Dekker, New York, 2001
  • S. Montgomery: Hopf algebras and their actions on rings, 2nd revised
    printing, Reg. Conf. Ser. Math., Vol. 82, Am. Math. Soc., Providence, 1997
  • D. E. Radford: Hopf algebras, Ser. Knots Everything, Vol. 49, World
    Scientific, Singapore, 2012