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In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.) Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism. As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is self-dual, so if one can define a dual of ''B'' (which is always possible if ''B'' is finite-dimensional), then it is automatically a bialgebra. == Formal definition == (''B'', ∇, η, Δ, ε) is a bialgebra over ''K'' if it has the following properties: * ''B'' is a vector space over ''K''; * there are ''K''-linear maps (multiplication) ∇: ''B'' ⊗ ''B'' → ''B'' (equivalent to ''K''-multilinear map ∇: ''B'' × ''B'' → ''B'') and (unit) η: ''K'' → ''B'', such that (''B'', ∇, η) is a unital associative algebra; * there are ''K''-linear maps (comultiplication) Δ: ''B'' → ''B'' ⊗ ''B'' and (counit) ε: ''B'' → ''K'', such that (''B'', Δ, ε) is a (counital coassociative) coalgebra; * compatibility conditions expressed by the following commutative diagrams: # Multiplication ∇ and comultiplication Δ #:: #: where τ: ''B'' ⊗ ''B'' → ''B'' ⊗ ''B'' is the linear map defined by τ(''x'' ⊗ ''y'') = ''y'' ⊗ ''x'' for all ''x'' and ''y'' in ''B'', # Multiplication ∇ and counit ε #:: # Comultiplication Δ and unit η #:: # Unit η and counit ε #:: 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「Bialgebra」の詳細全文を読む スポンサード リンク
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