|
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. ==Formal definition== Formally, a Hopf algebra is a (associative and coassociative) bialgebra ''H'' over a field ''K'' together with a ''K''-linear map ''S'': ''H'' → ''H'' (called the antipode) such that the following diagram commutes: Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless Sweedler notation, this property can also be expressed as : As for algebras, one can replace the underlying field ''K'' with a commutative ring ''R'' in the above definition.〔Underwood (2011) p.55〕 The definition of Hopf algebra is self-dual (as reflected in the symmetry of the above diagram), so if one can define a dual of ''H'' (which is always possible if ''H'' is finite-dimensional), then it is automatically a Hopf algebra.〔Underwood (2011) p.62〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hopf algebra」の詳細全文を読む スポンサード リンク
|