On groups of central type, non-degenerate and bijective cohomology classes

David, Nir Ben

doi:10.1007/s11856-009-0075-0

Cited by 13 publications

(8 citation statements)

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“…In [10] it is also proved that if H = {(a, φ(a) | a ∈ Fa n } is a group of I -type then the IYB group G = φ(Fa n ) naturally acts on the quotient group A = Fa n /K, where K = {a ∈ Fa n | φ(a) = 1} and one obtains a bijective associated 1-cocycle G → A with respect to this action. By a result of Etingof and Gelaki [9], this bijective 1-cocycle yields a non-degenerate 2-cocycle on the semi-direct product A G. This on its turn yields that A G is a group of central type in the sense of Ben David and Ginosar [3], i.e. a finite group H with a 2-cocycle c ∈ Z 2 (H, C * ) such that the twisted group algebra C c H is isomorphic to a full matrix algebra over the complex numbers.…”

Section: Involutive Yang-baxter Groupsmentioning

confidence: 88%

Semigroup Algebras and Noetherian Maximal Orders: a Survey

Jespers

2008

Acta Appl Math

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A survey is given on recent results describing when a semigroup algebra K[S] of a submonoid S of a polycyclic-by-finite group is a prime Noetherian maximal order. As an application one constructs concrete classes of finitely presented algebras that have the listed properties. Also some open problems are stated.Keywords Semigroup algebra · Noetherian · Maximal orderIn Sect. 2 we give necessary and sufficient conditions for a semigroup algebra K[S] of a submonoid S of a polycyclic-by-finite group to be a prime Noetherian maximal order. These results show that the problem can be reduced to S. The proofs rely on properties of minimal prime ideals of S and height one prime ideals of K [S]. These properties are being dealt with in Sect. 3. To obtain concrete examples, in Sect. 4 we deal with the Abelian case. These Abelian maximal orders are later used to show that some concrete non-commutative examples are maximal orders as well. Furthermore, in Sect. 5, we investigate a non-commutative class of examples, the monoids of I -type and their algebras. In order to construct such monoids one needs to describe finite solvable groups that act on finite Abelian groups of the same order and that satisfy some extra property. Such groups, called Involutive Yang-Baxter groups, are considered in Sect 6. Finally, in Sect. 7, the concrete class of the monoids of IG-type is studied. This class contains the monoids of I -type and provides many more examples of maximal orders. Furthermore, algebras K[S] of monoids of IG-type turn out to be epimorphic images of algebras K[T ] with T a monoid of I -type.Research partially supported by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish-Polish bilateral agreement BIL2005/VUB/06.

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Section: Involutive Yang-baxter Groupsmentioning

confidence: 88%

Semigroup Algebras and Noetherian Maximal Orders: a Survey

Jespers

2008

Acta Appl Math

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“…Hence, α(b 2 , c 2 , i 2 , j 2 ) = α(1, b −1 2 · c 2 , i 2 , j 2 ) (by (9), with a = b 2 ) = α(1, 1, i 2 , j 2 ) (by (7), with a = c −1 2 · b 2 ), and similarly for β(b 2 , c 2 , i 2 , j 2 ) = β(1, 1, i 2 , j 2 ). Note also that…”

Section: Generating All the Non-degenerate Solutions Associated To A mentioning

confidence: 94%

“…Hence it comes as no surprise that skew left braces, whose initial motivation was also the Yang-Baxter equation, have some connections with Hopf algebras too. Some of them are their relation with triangular semisimple and cosemisimple Hopf algebras explained in [24,9], and the relation with finite dimensional pointed Hopf algebras through rack theory, explained in [1] (we clarify the connection with rack theory in Section 5). These two classes of Hopf algebras have received a lot of attention recently since they are important in the program sketched in [2, pages 376, 377] to try to obtain a classification of finite-dimensional Hopf algebras.…”

Section: An Application To Hopf-galois Extensionsmentioning

confidence: 99%

Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks

Bachiller

2018

J. Knot Theory Ramifications

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Given a skew left brace B, a method is given to construct all the non-degenerate set-theoretic solutions (X, r) of the Yang Baxter equation such that the associated permutation group G(X, r) is isomorphic, as a skew left brace, to B. This method depends entirely on the brace structure of B. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.

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“…We call the triple (G, Z n , π) an (n-fold) I-datum 1 on the group G. It turns out that a 1-cocycle is bijective if and only if so are all the 1-cocycles in its cohomology class [4,Proposition 4.1]. The fact that bijectivity is a class property is respected by the cohomological structures in §3 and §4.…”

Section: Definitionsmentioning

confidence: 99%