霍普夫代數(shù)

內(nèi)容概要

　　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.

書籍目錄

PrefaceNotation1  Modules and algebras　1．Modules　2．Algebras over a commutative ring　3．Lie algebras　4．Semi-simple algebras　5．Finitely generated commutative algebras2 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 biaIgebras3 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 groups5  Applications to field theory　1．K／k—bialgebras　2．Jacobson's theorem　3．Modular extensionsAppendix：Categories and functors　A．1 Categories　A．2 Functors　A．3 Adjoint functors　A．4 Representable functors　A．5 φ-groups andφ-cogroupsReferencesIndex

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