In algebra, a measuring coalgebra of two algebras ''A'' and ''B'' is a coalgebra

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of the set of

homomorphisms In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

from ''A'' to ''B''. In other words, if coalgebras are thought of as a sort of linear analogue of sets, then the measuring coalgebra is a sort of linear analogue of the set of homomorphisms from ''A'' to ''B''. In particular its

group-like element In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. ...

s are (essentially) the homomorphisms from ''A'' to ''B''. Measuring coalgebras were introduced by .

Definition

A coalgebra ''C'' with a linear map from ''C''×''A'' to ''B'' is said to measure ''A'' to ''B'' if it preserves the algebra product and identity (in the coalgebra sense). If we think of the elements of ''C'' as linear maps from ''A'' to ''B'', this means that ''c''(''a''₁''a''₂) = Σ''c''₁(''a''₁)''c''₂(''a''₂) where Σ''c''₁⊗''c''₂ is the coproduct of ''c'', and ''c'' multiplies identities by the counit of ''c''. In particular if ''c'' is grouplike this just states that ''c'' is a homomorphism from ''A'' to ''B''. A measuring coalgebra is a universal coalgebra that measures ''A'' to ''B'' in the sense that any coalgebra that measures ''A'' to ''B'' can be mapped to it in a unique natural way.

Examples

*The group-like elements of a measuring coalgebra from ''A'' to ''B'' are the homomorphisms from ''A'' to ''B''. *The primitive elements of a measuring coalgebra from ''A'' to ''B'' are the derivations from ''A'' to ''B''. *If ''A'' is the algebra of continuous real functions on a compact Hausdorff space ''X'', and ''B'' is the real numbers, then the measuring coalgebra from ''A'' to ''B'' can be identified with finitely supported measures on ''X''. This may be the origin of the term "measuring coalgebra". *In the special case when ''A'' = ''B'', the measuring coalgebra has a natural structure of a Hopf algebra, called the Hopf algebra of the algebra ''A''.

References

* * *{{Citation , last1=Sweedler , first1=Moss E. , title=Hopf algebras , url=https://books.google.com/books?id=8FnvAAAAMAAJ , publisher=W. A. Benjamin, Inc., New York , series=Mathematics Lecture Note Series , year=1969 , mr=0252485 , zbl=0194.32901 Coalgebras