霍普夫代数

　　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.

暂缺《霍普夫代数》作者简介

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