Classifying bicrossed products of two Taft algebras

We classify all Hopf algebras which factor through two Taft algebras ${\mathbb{T}}_{n^2}(\stackrel{￣}{q})$ and respectively $T_{m^2}(q)$. To start with, all possible matched pairs between the two Taft algebras are described: if $\stackrel{￣}{q}\ne q^{n?1}$ then the matched pairs are in bijection with the group of d-th roots of unity in k, where $d=(m,n)$ while if $\stackrel{￣}{q}=q^{n?1}$ 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.     

Extending structures for Lie algebras

Let $\mathfrak{g }$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g }$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g }$ is a Lie subalgebra of $E$ . A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $\mathfrak{g }$ in $E$ : the unified product $\mathfrak{g } \,\natural \, V$ is associated to a system $(\triangleleft , \, \triangleright , \, f, \{-, \, -\})$ consisting of two actions $\triangleleft $ and $\triangleright $ , a generalized cocycle $f$ and a twisted Jacobi bracket $\{-, \, -\}$ on $V$ . There exists a Lie algebra structure $[-,-]$ on $E$ containing $\mathfrak{g }$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) \cong \mathfrak{g } \,\natural \, V$ . All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $\mathfrak{g }$ while the second object $\mathcal{H }^{2} (V, \mathfrak{g })$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ and $\mathcal{H }^{2} (V, \mathfrak{g })$ are worked out in detail in the case of flag extending structures.     

Classifying Bicrossed Products of Hopf Algebras

Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object ${\mathcal H}^{2} (A, H)$ . Equivalently, we classify up to a left A-linear Hopf algebra isomorphism, the set of all bicrossed products A???H associated to all possible matched pairs of Hopf algebras $(A, H, \triangleleft, \triangleright)$ that can be defined between A and H. In the construction of ${\mathcal H}^{2} (A, H)$ the key role is played by special elements of $CoZ^{1} (H, A) \times {\rm Aut}\,_{\rm CoAlg}^1 (H)$ , where CoZ ¹ (H, A) is the group of unitary cocentral maps and ${\rm Aut}\,_{\rm CoAlg}^1 (H)$ is the group of unitary automorphisms of the coalgebra H. Among several applications and examples, all bicrossed products H ₄???k[C _n ] are described by generators and relations and classified: they are quantum groups at roots of unity H _{4n, ω} which are classified by pure arithmetic properties of the ring ? _n . The Dirichlet’s theorem on primes is used to count the number of types of isomorphisms of this family of 4n-dimensional quantum groups. As a consequence of our approach the group Aut _Hopf(H _{4n, ω} ) of Hopf algebra automorphisms is fully described.     

Bicrossed products with the Taft algebra

Archiv der Mathematik - 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...     

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 → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H _α⋈β G ≅H _{α′⋈β′} (G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H _α⋈β G ≅ H _{α′⋈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 products of groups are given.     

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.     

Braidings on the Category of Bimodules,Azumaya Algebras and Epimorphisms of Rings

Let A be an algebra over a commutative ring k. We prove that braidings on the category of A-bimodules are in bijective correspondence to canonical R-matrices, these are elements in A???A???A satisfying certain axioms. We show that all braidings are symmetries. If A is commutative, then there exists a braiding on ${}_A\mathcal{M}_A$ if and only if k→A is an epimorphism in the category of rings, and then the corresponding R-matrix is trivial. If the invariants functor $G = (-)^A:\ {}_A\mathcal{M}_A\to \mathcal{M}_k$ is separable, then A admits a canonical R-matrix; in particular, any Azumaya algebra admits a canonical R-matrix. Working over a field, we find a remarkable new characterization of central simple algebras: these are precisely the finite dimensional algebras that admit a canonical R-matrix. Canonical R-matrices give rise to a new class of examples of simultaneous solutions for the quantum Yang–Baxter equation and the braid equation.