Editing Hopf algebra (section)

{{Short description|Construction in algebra}}
{{Use American English|date=January 2019}}
In [[mathematics]], a '''Hopf algebra''', named after [[Heinz Hopf]], is a structure that is simultaneously a ([[unital algebra|unital]] associative) [[Associative algebra|algebra]] and a (counital coassociative) [[coalgebra]], with these structures' compatibility making it a [[bialgebra]], and that moreover is equipped with an [[antihomomorphism]] satisfying a certain property. The [[representation theory]] of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.

Hopf algebras occur naturally in [[algebraic topology]], where they originated and are related to the [[H-space]] concept, in [[group scheme]] theory, in [[group theory]] (via the concept of a [[group ring]]), and in numerous other places, making them probably the most familiar type of [[bialgebra]]. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. They have diverse applications ranging from [[condensed matter physics]] and [[quantum field theory]]<ref>{{cite journal | last1 = Haldane | first1 = F. D. M. | last2 = Ha | first2 = Z. N. C. | last3 = Talstra | first3 = J. C. | last4 = Bernard | first4 = D. | last5 = Pasquier | first5 = V. | year = 1992 | title = Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory | journal = Physical Review Letters | volume = 69 | issue = 14| pages = 2021–2025 | doi=10.1103/physrevlett.69.2021 | pmid=10046379| bibcode = 1992PhRvL..69.2021H}}</ref> to [[string theory]]<ref>{{cite journal | last1 = Plefka | first1 = J. | last2 = Spill | first2 = F. | last3 = Torrielli | first3 = A. | year = 2006 | title = Hopf algebra structure of the AdS/CFT S-matrix | journal = Physical Review D | volume = 74 | issue = 6| page = 066008 | doi = 10.1103/PhysRevD.74.066008 | arxiv = hep-th/0608038 | bibcode = 2006PhRvD..74f6008P| s2cid = 2370323 }}</ref> and [[Large Hadron Collider|LHC phenomenology]].<ref>{{Cite journal|last1=Abreu|first1=Samuel|last2=Britto|first2=Ruth|author2-link= Ruth Britto |last3=Duhr|first3=Claude|last4=Gardi|first4=Einan|date=2017-12-01|title=Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case|journal=Journal of High Energy Physics|language=en|volume=2017|issue=12|pages=90|doi=10.1007/jhep12(2017)090|issn=1029-8479|arxiv=1704.07931|bibcode=2017JHEP...12..090A|s2cid=54981897}}</ref>