# Measuring coalgebra

In algebra, a **measuring coalgebra** of two algebras *A* and *B* is a coalgebra enrichment of the set of homomorphisms 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 elements are (essentially) the homomorphisms from *A* to *B*. Measuring coalgebras were introduced by Sweedler (1968, 1969).

## DefinitionEdit

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.

## ExamplesEdit

- 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*.

## ReferencesEdit

- Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010),
*Algebras, rings and modules. Lie algebras and Hopf algebras*, Mathematical Surveys and Monographs,**168**, Providence, RI: American Mathematical Society, ISBN 978-0-8218-5262-0, MR 2724822, Zbl 1211.16023 - Sweedler, Moss E. (1968), "The Hopf algebra of an algebra applied to field theory",
*J. Algebra*,**8**: 262–276, doi:10.1016/0021-8693(68)90059-8, MR 0222053 - Sweedler, Moss E. (1969),
*Hopf algebras*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, MR 0252485, Zbl 0194.32901