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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

If for instance， we replace the finite group G in the above argumentby a topological group and k by the field of real numbers or the field ofcomplex numbers， or if we take G to be an algebraic group over analgebraically closed field k and A is replaced by the k-algebra of allcontinuous representative functions or of all regular functions over G，then A turns out to be a k-Hopf algebra in exactly the same manner.These algebraic systems play an important role when studying thestructure of G. Similarly， a k-Hopf algebra structure can be definednaturally on the universal enveloping algebra of a k-Lie algebra.The universal enveloping algebra of the Lie algebra of asemi-simple algebraic group turns out to be （in a sense） the dual oftheHopf algebra defined above. These constitute some of the mostnatural examples of Hopf algebras. The general structure of suchalgebraic systems has recently become a focus of interest in con-junction with its applications to the theory of algebraic groups or theGalois theory of purely inseparable extensions， and a great deal ofresearch is currently being conducted in this area.