In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) 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[1] to string theory[2] and LHC phenomenology.[3]
^Haldane, F. D. M.; Ha, Z. N. C.; Talstra, J. C.; Bernard, D.; Pasquier, V. (1992). "Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory". Physical Review Letters. 69 (14): 2021–2025. Bibcode:1992PhRvL..69.2021H. doi:10.1103/physrevlett.69.2021. PMID 10046379.
^Plefka, J.; Spill, F.; Torrielli, A. (2006). "Hopf algebra structure of the AdS/CFT S-matrix". Physical Review D. 74 (6): 066008. arXiv:hep-th/0608038. Bibcode:2006PhRvD..74f6008P. doi:10.1103/PhysRevD.74.066008. S2CID 2370323.
^Abreu, Samuel; Britto, Ruth; Duhr, Claude; Gardi, Einan (2017-12-01). "Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case". Journal of High Energy Physics. 2017 (12): 90. arXiv:1704.07931. Bibcode:2017JHEP...12..090A. doi:10.1007/jhep12(2017)090. ISSN 1029-8479. S2CID 54981897.
In mathematics, a Hopfalgebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative)...
associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A Hopfalgebra is a...
noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopfalgebras), compact matrix...
abstract algebra, a representation of a Hopfalgebra is a representation of its underlying associative algebra. That is, a representation of a Hopfalgebra H...
In mathematics, a Hopfalgebra, H, is quasitriangular if there exists an invertible element, R, of H ⊗ H {\displaystyle H\otimes H} such that R Δ ( x...
as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopfalgebra; in this case...
be extended by giving an antipode to create a Hopfalgebra structure. Note: In this article, all algebras are assumed to be unital and associative. The...
both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras...
In mathematics, a shuffle algebra is a Hopfalgebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢...
In algebra, the Pareigis Hopfalgebra is the Hopfalgebra over a field k whose left comodules are essentially the same as complexes over k, in the sense...
the group Hopfalgebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopfalgebras are foundational...
algebra A p ∗ {\displaystyle {\mathcal {A}}_{p}^{*}} (also denoted A ∗ {\displaystyle {\mathcal {A}}^{*}} ) is a graded noncommutative Hopfalgebra which...
prime number p {\displaystyle p} , the Steenrod algebra A p {\displaystyle A_{p}} is the graded Hopfalgebra over the field F p {\displaystyle \mathbb {F}...
In algebra, the Malvenuto–Poirier–Reutenauer Hopfalgebra of permutations or MPR Hopfalgebra is a Hopfalgebra with a basis of all elements of all the...
finite-dimensional Hopfalgebra, by a 1969 theorem of Larson-Sweedler on Hopf modules and integrals. The direct product and tensor product of Frobenius algebras are...
In algebra, a Taft Hopfalgebra is a Hopfalgebra introduced by Earl Taft (1971) that is neither commutative nor cocommutative and has an antipode of large...
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal...
In mathematics, a braided Hopfalgebra is a Hopfalgebra in a braided monoidal category. The most common braided Hopfalgebras are objects in a Yetter–Drinfeld...
Hopfalgebra ( A , ∇ , η , Δ , ε , S , R , ν ) {\displaystyle (A,\nabla ,\eta ,\Delta ,\varepsilon ,S,{\mathcal {R}},\nu )} is a quasitriangular Hopf...
algebra can be given the structure of a Hopfalgebra. See Tensor algebra for details. The symmetric algebra S(V) is the universal enveloping algebra of...
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure...
symmetric algebra) inherits a bialgebra structure, and, indeed, a Hopfalgebra structure, from the tensor algebra. See the article on tensor algebras for a...
a Hopfalgebra do form a group. A primitive element is an element x that satisfies Δ(x) = x ⊗ 1 + 1 ⊗ x. The primitive elements of a Hopfalgebra form...
Global Information