It has only been since the late 1960s that Hopfalgebras, as algebraic systems, became objects of study from an algebraic standpoint.However, beginning with the research on representation theory through the use of the representative rings of Lie groups by Hochschild-Mostow, Hopf algebras have been taken up extensively as algebraic systems and also used in applications.

Preface

Notation

1 Modules and algebras

　1．Modules

　2．Algebras over a commutative ring

　3．Lie algebras

　4．Semi-simple algebras

　5．Finitely generated commutative algebras

2 Hopf algebras

 1．Bialgcbras and Hopf algebras

 2．The representative bialgebras of semigroups

 3．The duality between algebras and coalgebras

 4．Irreducible bialgebras　

 5．Irreducible cocommutative biaIgebras

3 Hopr algebras and relnmmamtlom of group

　1．Comodules and bimodules

　2．Bimodules and biaIgebms

　3．Integrals for Hopf algebras

　4．The duality theorem　

4 ApplimlJons to algebraic groups　

 1．Affme k-varieties　

 2．Atone k-groups

 3．Lie algebras of affme algebraic k-groups

 4．Factor groups

 5．Unipotent groups and solvable groups

 6．Completely reducible groups

5 Applications to field theory

　1．K／k—bialgebras

　2．Jacobson's theorem

　3．Modular extensions

Appendix：Categories and functors

　A．1 Categories

　A．2 Functors

　A．3 Adjoint functors

　A．4 Representable functors

　A．5 φ-groups andφ-cogroups

References

Index