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Sweedler's Hopf Algebra
In mathematics, introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra ''H''4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative. Definition The following infinite dimensional Hopf algebra was introduced by . The Hopf algebra is generated as an algebra by three elements ''x'', ''g'' and ''g''-1. The coproduct Δ is given by :Δ(g) = ''g'' ⊗''g'', Δ(''x'') = 1⊗''x'' + ''x'' ⊗''g'' The antipode ''S'' is given by :''S''(''x'') = –''x'' ''g''−1, ''S''(''g'') = ''g''−1 The counit ε is given by :ε(''x'')=0, ε(''g'') = 1 Sweedler's 4-dimensional Hopf algebra ''H''4 is the quotient of this by the relations :''x''2 = 0, ''g''2 = 1, ''gx'' = –''xg'' so it has a basis 1, ''x'', ''g'', ''xg'' . Note that Montgomery describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on ''H''4⊗''H''4. This Hopf algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebra
Hopf is a German surname. Notable people with the surname include: *Eberhard Hopf (1902–1983), Austrian mathematician *Hans Hopf (1916–1993), German tenor *Heinz Hopf (1894–1971), German mathematician *Heinz Hopf (actor) (1934–2001), Swedish actor *Ludwig Hopf (1884–1939), German physicist *Maria Hopf Maria Hopf (13 September 1913 – 24 August 2008) was a pioneering archaeobotanist, based at the RGZM, Mainz. Career Hopf studied botany from 1941–44, receiving her doctorate in 1947 on the subject of soil microbes. She then worked in phyto ... (1914-2008), German botanist and archaeologist {{surname, Hopf German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareigis Hopf Algebra
In algebra, the Pareigis Hopf algebra is the Hopf algebra over a field ''k'' whose left comodules are essentially the same as complexes over ''k'', in the sense that the corresponding monoidal categories are isomorphic. It was introduced by as a natural example of a Hopf algebra that is neither commutative nor cocommutative. Construction As an algebra over ''k'', the Pareigis algebra is generated by elements ''x'',''y'', 1/''y'', with the relations ''xy'' + ''yx'' = ''x''2 = 0. The coproduct takes ''x'' to ''x''⊗1 + (1/''y'')⊗''x'' and ''y'' to ''y''⊗''y'', and the counit takes ''x'' to 0 and ''y'' to 1. The antipode takes ''x'' to ''xy'' and ''y'' to its inverse and has order 4. Relation to complexes If ''M'' = ⊕''M''''n'' is a complex with differential ''d'' of degree –1, then ''M'' can be made into a comodule over ''H'' by letting the coproduct take ''m'' to Σ ''y''''n''⊗''m''''n'' + ''y''''n''+1''x''⊗''dm''''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Algebra
''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief. In 2004, ''Journal of Algebra'' announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to computational algebra was vol. 292, no. 1 (October 2005). The Editor-in-Chief of the ''Journal of Algebra'' is Michel Broué, Université Paris Diderot, and Gerhard Hiß, Rheinisch-Westfälische Technische Hochschule Aachen ( RWTH) is Editor of the computational algebra section. See also *Susan Montgomery M. Susan Montgomery (born 2 April 1943 in Lansing, MI) is a distinguished American mathematician whose current research interests concern noncommutative algebras: in parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |