Bicrossed Products for Finite Groups

Agore, A. L.; Chirvăsitu, Alexandru; Ion, Bogdan; Militaru, Gigel

doi:10.1007/s10468-009-9145-6

Cited by 17 publications

(23 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, if ψ −1 is the inverse of ψ then using the same arguments as above one can easily prove that there exists a unitary coalgebra map r : K[C n ] → T m 2 (q) and a Hopf algebra map v : K[C n ] → K[C n ] such that for any z ∈ K[C n ] we have ψ −1 (1#z) = r(z (1) )#v(z (2) ). This gives 1#z = ψ • ψ −1 (1#z) = u r z (1) r v z (2) #v v z (3) .…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Hopf Algebras which Factorize through the Taft Algebra Tm2(q) and the Group Hopf Algebra K[Cn]

Agore¹

2018

SIGMA

View full text Add to dashboard Cite

Abstract. We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra T m 2 (q) and the group Hopf algebra K[C n ]: they are nm 2 -dimensional quantum groups T ω nm 2 (q) associated to an n-th root of unity ω. Furthermore, using Dirichlet's prime number theorem we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if d = gcd(m, ν(n)) andis the prime decomposition ofd then the number of types of Hopf algebras that factorize through T m 2 (q) and K[C n ] is equal to (α 1 + 1)(α 2 + 1) · · · (α r + 1), where ν(n) is the order of the group of n-th roots of unity in K. As a consequence of our approach, the automorphism groups of these Hopf algebras are described as well.

show abstract

Section: Resultsmentioning

confidence: 99%

“…The factorization problem was considered in many settings ranging from groups [3,19,21] to (co)algebras [8,7,10], Lie algebras [14,17], locally compact quantum groups [22] or fusion categories [9]. We refer the reader to [2] and the references therein for a thorough overview on recent developments.…”

Section: Introductionmentioning

confidence: 99%

Hopf Algebras which Factorize through the Taft Algebra Tm2(q) and the Group Hopf Algebra K[Cn]

Agore¹

2018

SIGMA

View full text Add to dashboard Cite

show abstract

“…turns out to be very difficult. Probably the most conclusing argument comes from the group case where the description and classification of all bicrossed products between two finite cyclic groups is still an open question (see [4] for an up-to-date account on the subject). Nevertheless, this topic has been intensively studied in several papers including [1], [3], [7], [8], [9], [14], [22], to mention only a few, and it turned out to be an effective method of constructing new objects (groups, associative algebras, Hopf algebras etc.)…”

Section: Introductionmentioning

confidence: 99%

Bicrossed products with the Taft algebra

Agore

Nastasescu

2019

Arch. Math.

Self Cite

View full text Add to dashboard Cite

Let G be a group which admits a generating set consisting of finite order elements. We prove that any Hopf algebra which factorizes through the Taft algebra and the group Hopf algebra K[G] (equivalently, any bicrossed product between the aforementioned Hopf algebras) is isomorphic to a smash product between the same two Hopf algebras. The classification of these smash products is shown to be strongly linked to the problem of describing the group automorphisms of G. As an application, we completely describe by generators and relations and classify all bicrossed products between the Taft algebra and the group Hopf algebra K[D 2n ], where D 2n denotes the dihedral groups.2010 Mathematics Subject Classification. 16T05, 16S40.

show abstract

“…However, although the statement of the problem is very simple and natural, no major progress has been made so far as we still lack exhaustive methods to tackle it. For instance, even the description and classification of groups which factor through two finite cyclic groups is still an open problem although there are several papers dealing with it, such as the four papers by J. Douglas [8] and the more recent one [1] which solves completely the problem in the special case where one of the groups is of prime order.…”

Section: Introductionmentioning

confidence: 99%

Classifying bicrossed products of two Taft algebras

Agore

2018

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

We classify all Hopf algebras which factor through two Taft algebras T n 2 (q) and respectively T m 2 (q). To start with, all possible matched pairs between the two Taft algebras are described: if q = q n−1 then the matched pairs are in bijection with the group of d-th roots of unity in k, where d = (m, n) while if q = q n−1 then besides the matched pairs above we obtain an additional family of matched pairs indexed by k * . The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups.2010 Mathematics Subject Classification. 16T05, 16S40.

show abstract

Bicrossed Products for Finite Groups

Cited by 17 publications

References 12 publications

Hopf Algebras which Factorize through the Taft Algebra T<sub>m<sup>2</sup></sub>(q) and the Group Hopf Algebra K[C<sub>n</sub>]

Hopf Algebras which Factorize through the Taft Algebra T<sub>m<sup>2</sup></sub>(q) and the Group Hopf Algebra K[C<sub>n</sub>]

Bicrossed products with the Taft algebra

Classifying bicrossed products of two Taft algebras

Contact Info

Product

Resources

About