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 {{harvs|txt|authorlink=Moss Sweedler|last=Sweedler|year1=1968|year2=1969}}.
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(a1a2) = Σc1(a1)c2(a2) where Σc1⊗c2 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 | mr=2724822 | zbl=1211.16023
|last1=Hazewinkel|first1= Michiel|last2= Gubareni|first2= Nadiya|last3= Kirichenko|first3= V. V.
|title=Algebras, rings and modules. Lie algebras and Hopf algebras|series= Mathematical Surveys and Monographs|volume= 168|publisher= American Mathematical Society|place= Providence, RI|year= 2010|isbn= 978-0-8218-5262-0 }}
- {{citation|mr=0222053
|last=Sweedler|first= Moss E.
|title=The Hopf algebra of an algebra applied to field theory
|journal=J. Algebra|volume= 8 |year=1968 |issue=3 |pages=262–276|doi=10.1016/0021-8693(68)90059-8|doi-access=}}
- {{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 | isbn=9780805392548 | mr=0252485 | zbl=0194.32901 }}