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.

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