Constructing Hopf braces

We investigate Hopf braces, a concept recently introduced by Angiono, Galindo and Vendramin in connection to the quantum Yang-Baxter equation. More precisely, we propose two methods for constructing Hopf braces. The first one uses matched pairs of Hopf algebras while the second one relies on category theoretic tools

Classifying complements for associative algebras

For a given extension $A \subset E$ of associative algebras we describe and classify up to an isomorphism all $A$-complements of $E$, i.e. all subalgebras $X$ of $E$ such that $E = A + X$ and $A \cap X = \{0\}$. Let $X$ be a given complement and $(A, \, X, \, \triangleright, \triangleleft, \leftharpoonup, \rightharpoonup \bigl)$ the canonical matched pair associated with the factorization $E = A + X$. We introduce a new type of deformation of the algebra $X$ by means of the given matched pair and prove that all $A$-complements of $E$ are isomorphic to such a deformation of $X$. Several explicit examples involving the matrix algebra are provided.Comment: 10 pages; to appear in Linear Algebra and its Application

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.Comment: to appear in Communications in Algebr

Coquasitriangular structures for extensions of Hopf algebras. Applications

Let $A \subseteq E$ be an extension of Hopf algebras such that there exists a normal left $A$-module coalgebra map $\pi : E \to A$ that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra $E$ in terms of the datum $(A, E, \pi)$ as follows: first, any such extension $E$ is isomorphic to a unified product $A \ltimes H$, for some unitary subcoalgebra $H$ of $E$ (\cite{am2}). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product $A \ltimes H$ and a certain set of datum $(p, \tau, u, v)$ related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite dimensional quantum double $D_{\lambda}(A, H) = A \bowtie_{\tau} H$ to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.Comment: 16 pages, to appear in Glasgow Math.

The maximal dimension of unital subalgebras of the matrix algebra

Using Wederburn's main theorem and a result of Gerstenhaber we prove that, over a field of characteristic zero, the maximal dimension of a proper unital subalgebra in the $n \times n$ matrix algebra is $n^2 - n + 1$ and furthermore this upper bound is attained for the so-called parabolic subalgebras. We also investigate the corresponding notion of parabolic coideals for matrix coalgebras and prove that the minimal dimension of a non-zero coideal of the matrix coalgebra ${\mathcal M}^n (k)$ is $n-1$.Comment: 7 pages; title changed. Final version to appear in Forum Math. DOI:10.1515/forum-2015-024

Free Poisson Hopf algebras generated by coalgebras

We construct the analogue of Takeuchi's free Hopf algebra in the setting of Poisson Hopf algebras. More precisely, we prove that there exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the category of Poisson Hopf algebras to the category of coalgebras has a left adjoint. In particular, we also prove that the category of Poisson Hopf algebras is a reflective subcategory of the category of Poisson bialgebras. Along the way, we describe coproducts and coequalizers in the category of Poisson Hopf algebras, therefore showing that the latter category is cocomplete.Comment: to appear in J. Math. Phy

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.Comment: Minor changes from previous version. To appear in Proceedings of the American Mathematical Societ

Classifying bicrossed products of two Taft algebras

We classify all Hopf algebras which factorize through two Taft algebras $\mathbb{T}_{n^{2}}(\bar{q})$ and respectively $T_{m^{2}}(q)$. To start with, all possible matched pairs between the two Taft algebras are described: if $\bar{q} \neq q^{n-1}$ then the matched pairs are in bijection with the group of $d$-th roots of unity in $k$, where $d = (m,\,n)$ while if $\bar{q} = q^{n-1}$ 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.Comment: Continues arXiv:1205.6110, arXiv:1205.6564; restates preliminaries and definitions for sake of clarity. Final version, to appear in J. Pure Appl. Algebr

Crossed product of Hopf algebras

The main properties of the crossed product in the category of Hopf algebras are investigated. Let $A$ and $H$ be two Hopf algebras connected by two morphism of coalgebras \triangleright : H\ot A \to A, f:H\ot H\to A. The crossed product A #_{f}^{\triangleright} H is a new Hopf algebra containing $A$ as a normal Hopf subalgebra. Furthermore, a Hopf algebra $E$ is isomorphic as a Hopf algebra to a crossed product of Hopf algebras A #_{f}^{\triangleright} H if and only if $E$ factorizes through a normal Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1_{E} \in H$. The universality of the construction, the existence of integrals, commutativity or involutivity of the crossed product are studied. Looking at the quantum side of the construction we shall give necessary and sufficient conditions for a crossed product to be a coquasitriangular Hopf algebra. In particular, all braided structures on the monoidal category of A #_{f}^{\triangleright} H-comodules are explicitly described in terms of their components. As an example, the braidings on a crossed product between $H_{4}$ and $k[C_{3}]$ are described in detail.Comment: 23 pages, to appear in Comm. Algebr

It\^o's theorem and metabelian Leibniz algebras

We prove that the celebrated It\^{o}'s theorem for groups remains valid at the level of Leibniz algebras: if $\mathfrak{g}$ is a Leibniz algebra such that $\mathfrak{g} = A + B$, for two abelian subalgebras $A$ and $B$, then $\mathfrak{g}$ is metabelian, i.e. $[ \, [\mathfrak{g}, \, \mathfrak{g}], \, [ \mathfrak{g}, \, \mathfrak{g} ] \, ] = 0$. A structure type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimension $1$ are described, classified and their automorphisms groups are explicitly determined as subgroups of a semidirect product of groups $P^* \ltimes \bigl(k^* \times {\rm Aut}_{k} (P) \bigl)$ associated to any vector space $P$.Comment: Final version; to appear in Linear Multilinear Algebr