Hopf algebras and combinatorics

Nichols, Warren D.; Sweedler, Moss E.

doi:10.1090/conm/006/646799

Cited by 41 publications

(5 citation statements)

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“…Example 3.9. [NS82] Start with the Z-module C " Z 8Z ' Z 2Z . Let x " p1, 0q and z " p0, 1q and endow C with a coalgebra structure by setting ∆pxq " 0 and ∆pzq " 4x b x .…”

Section: Product Of Submodulesmentioning

confidence: 99%

Pathlike Co/bialgebras and their antipodes with applications to bi- and Hopf algebras appearing in topology, number theory and physics

Kaufmann,

2021

Preprint

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We develop an algebraic theory of flavored and pathlike co-, bi-and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in topology, number theory and physics. In particular, we can precisely give conditions the invertibility of characters needed for renormalization in the formulation of Connes and Kreimer and in the canonical examples these conditions are met. To construct the antipodes directly, we discuss formal localization constructions and quantum deformations. These allow to construct and naturally explain the appearance Brown style coactions. Using previous results, we can tie all the relevant coalgebras to categorical constructions and the bialgebra structures to Feynman categories.

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“…Example 3.9. [NS82] Start with the Z-module C " Z 8Z ' Z 2Z . Let x " p1, 0q and z " p0, 1q and endow C with a coalgebra structure by setting ∆pxq " 0 and ∆pzq " 4x b x .…”

Section: Product Of Submodulesmentioning

confidence: 99%

Pathlike Co/bialgebras and their antipodes with applications to bi- and Hopf algebras appearing in topology, number theory and physics

Kaufmann,

2021

Preprint

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“…We are merely claiming that with the original algebra structure it is possible to find a coproduct on ⊕Z n that preserves the sub-algebra spanned by descent sums and thus gives a bialgebra structure on the Solomon descent algebra that is a subbialgebra of ⊕Z n . We also draw the reader's attention to the fact that the issues highlighted in [3], about the difficulties in putting a bialgebra structure on a sub-object of a bialgebra, are circumvented by the very explicit nature of our constructions.…”

Section: Permutation Bialgebrasmentioning

confidence: 99%

Group Bialgebras and Permutation Bialgebras

Crossley

2013

Glasgow Math. J.

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Malvenuto and Reutenauer (C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), 967-982) showed how the total symmetric group ring ⊕ n Z n could be made into a Hopf algebra with a very nice structure which admitted the Solomon descent algebra as a sub-Hopf algebra. To do this they replaced the group multiplication by a convolution product, thus distancing their structure from the group structure of n . In this paper we examine what is possible if we keep to the group multiplication, and we also consider the question for more general families of groups. We show that a Hopf algebra structure is not possible, but cocommutative and non-cocommutative counital bialgebras can be obtained, arising from certain diagrams of group homomorphisms. In the case of the symmetric groups we note that all such structures are weak in the sense that the dual algebras have many zero-divisors, but structures which respect descent sums can be found.

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“…Other useful references are [7,26,30,38,48]. Hopf algebras in the context of combinatorics appeared in the work of Rota [42], and Joni and Rota [28], see also [25,41,46,47].…”

Section: Introductionmentioning

confidence: 99%

On matrix differential equations in the Hopf algebra of renormalization

Ebrahimi‐Fard¹,

Manchon²

2006

Advances in Theoretical and Mathematical Physics

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We establish Sakakibara's differential equations [M. Sakakibara, On the differential equations of the characters for the renormalization group, Mod. Phys. Lett. A 19 (2004), 1453.] in a matrix setting for the counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff decomposition in any connected graded Hopf algebra, thus including Feynman rules in perturbative renormalization as a key example.

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Hopf algebras and combinatorics

Cited by 41 publications

References 0 publications

Pathlike Co/bialgebras and their antipodes with applications to bi- and Hopf algebras appearing in topology, number theory and physics

Pathlike Co/bialgebras and their antipodes with applications to bi- and Hopf algebras appearing in topology, number theory and physics

Group Bialgebras and Permutation Bialgebras

On matrix differential equations in the Hopf algebra of renormalization

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