In algebra, a measuring coalgebra of two algebras A and B is a coalgebraenrichment 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(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.
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, vol. 168, Providence, RI: American Mathematical Society, ISBN978-0-8218-5262-0, MR 2724822, Zbl 1211.16023
Sweedler, Moss E. (1968), "The Hopf algebra of an algebra applied to field theory", J. Algebra, 8 (3): 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, ISBN9780805392548, MR 0252485, Zbl 0194.32901
December 26, 2023
measuring, coalgebra, algebra, measuring, coalgebra, algebras, coalgebra, enrichment, homomorphisms, from, other, words, coalgebras, thought, sort, linear, analogue, sets, then, measuring, coalgebra, sort, linear, analogue, homomorphisms, from, particular, gro. 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 Definition editA 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 Sc1 a1 c2 a2 where Sc1 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 editThe 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 editHazewinkel Michiel Gubareni Nadiya Kirichenko V V 2010 Algebras rings and modules Lie algebras and Hopf algebras Mathematical Surveys and Monographs vol 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 3 262 276 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 ISBN 9780805392548 MR 0252485 Zbl 0194 32901 Retrieved from https en wikipedia org w index php title Measuring coalgebra amp oldid 1188050771, wikipedia, wiki, book, books, library,
