arXiv subjects

A. L. Agore

Publications and source records attributed to A. L. Agore.

43 records · Page 3Linked to original sources

arXiv2011

Schreier type theorems for bicrossed products

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, α, β)$ is deformed using a combinatorial datum $(σ, v, r)$ consisting of an automorphism $σ$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), α', β' \bigl)$ such that there exist an $σ$-invariant isomorphism of groups $H {}_α \bowtie_β G \cong H {}_{α'} \bowtie_{β'} (G,*)$. Moreover, if we fix the group $H$ and the automorphism $σ\in \Aut(H)$ then any $σ$-invariant isomorphism $H {}_α \bowtie_β G \cong H {}_{α'} \bowtie_{β'} G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.

math.GR↗

arXiv2011

Extending Structures II: The Quantum Version

Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E with 1_{E} \in H and the multiplication map $A\otimes H \to E$ is bijective. The tool we use is a new product, we call it the unified product, in the construction of which A and H are connected by three coalgebra maps: two actions and a generalized cocycle. Both the crossed product of an Hopf algebra acting on an algebra and the bicrossed product of two Hopf algebras are special cases of the unified product. A Hopf algebra E factorizes through A and H if and only if E is isomorphic to a unified product of A and H. All such Hopf algebras E are classified up to an isomorphism that stabilizes A and H by a Schreier type classification theorem. A coalgebra version of lazy 1-cocycles as defined by Bichon and Kassel plays the key role in the classification theorem.

math.RA↗

arXiv2010

Limits of Coalgebras, Bialgebras and Hopf Algebras

We give the explicit construction of the product of an arbitrary family of coalgebras, bialgebras and Hopf algebras: it turns out that the product of an arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family of coalgebras (resp. bialgebras, Hopf algebras). The equalizers of two morphisms of coalgebras (resp. bialgebras, Hopf algebras) are also described explicitly. As a consequence the categories of coalgebras, bialgebras and Hopf algebras are shown to be complete and a explicit description for limits in the above categories is given.

math.QA↗

arXiv2010

Categorical Constructions for Hopf Algebras

We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a Hopf coreflection and on every algebra there exists a cofree Hopf algebra. In this way we give an affirmative answer to a forty years old problem posed by Sweedler. On the route the coequalizers and the coproducts in the category of Hopf algebras are explicitly described.

math.QA↗

arXiv2010

Monomorphisms of Coalgebras

We prove new necessary and sufficient conditions for a morphism of coalgebras to be a monomorphism, different from the ones already available in the literature. More precisely, $ϕ: C \to D$ is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide if and only if $\sum_{i \in I}ε(a^{i})b^{i} = \sum_{i \in I} a^{i} ε(b^{i})$, for all $\sum_{i \in I}a^{i} \otimes b^{i} \in C \square_{D} C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.

math.RA↗

arXiv2009

Bicrossed products for finite groups

We investigate one question regarding bicrossed products of finite groups which we believe has the potential of being approachable for other classes of algebraic objects (algebras, Hopf algebras). The problem is to classify the groups that can be written as bicrossed products between groups of fixed isomorphism types. The groups obtained as bicrossed products of two finite cyclic groups, one being of prime order, are described.

math.GR↗

arXiv2008

Crossed product of groups. Applications

We survey the extensions of a group by a group using crossed products instead of exact sequences of groups. The approach has various advantages, one of them being that the crossed product is an universal object. Several new applications are given, a general Schreier type theorem is proved and a few open problems are left.

math.GR↗