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We introduce a new variety of set-theoretic non-associative algebras, P{\l}onka bi-magmas, to describe and classify all solutions of the set-theoretic Yang-Baxter (YB) equation of Baaj-Long-Skandalis (BLS) type. We also study new classes of YB-solutions (bi-connected, simple), and classify the BLS-solutions that fit into those classes. There are only countably many isomorphism classes of simple BLS-solutions, for instance, and they are all describable in terms of the odometer transformations familiar from ergodic theory. Placing Drinfel'd's problem of classifying the set-theoretic solutions of the Yang-Baxter equation in a universal-algebra context with a combinatorial flavor, we also prove the existence of adjunctions between various categories of solutions., Comment: 45 pages + references

We provide an answer, in a special but relevant case, to an open problem of Drinfel'd by proving that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, nondegenerate, surjective, finite order, unitary or indecomposable solutions of FS type are classified. For instance, if $|X| = n$, then the number of isomorphism classes of all such solutions on $X$ that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-nondegenerate is: (a) the Davis number $d(n)$, (b) $\sum_{m|n} \, p(m)$, where $p(m)$ is the Euler partition number, (c) $\tau(n) + \sum_{d|n}\left\lfloor \frac d2\right\rfloor$, where $\tau(n)$ is the number of divisors of $n$, or (d) the Harary number $\mathfrak{c} (n)$. The automorphism groups of such solutions can also be recovered as automorphism groups $\mathrm{Aut}(f)$ of sets $X$ equipped with a single endo-function $f:X\to X$. We describe all groups of the form $\mathrm{Aut}(f)$ as iterations of direct and (possibly infinite) wreath products of cyclic or full symmetric groups, characterize the abelian ones as products of cyclic groups, and produce examples of symmetry groups of FS solutions not of the form $\mathrm{Aut}(f)$., Comment: minor phrasing changes following comments; 47 pages + references

We introduce the \emph{universal algebra} of two Poisson algebras $P$ and $Q$ as a commutative algebra $A:={\mathcal P} (P, \, Q )$ satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra $P$ and several of its applications are highlighted. For any Poisson $P$-module $U$, we construct a functor $U \otimes - \colon {}_{A} {\mathcal M} \to {}_Q{\mathcal P}{\mathcal M}$ from the category of $A$-modules to the category of Poisson $Q$-modules which has a left adjoint whenever $U$ is finite dimensional. Similarly, if $V$ is an $A$-module, then there exists another functor $ - \otimes V \colon {}_P{\mathcal P}{\mathcal M} \to {}_Q{\mathcal P}{\mathcal M}$ connecting the categories of Poisson representations of $P$ and $Q$ and the latter functor also admits a left adjoint if $V$ is finite dimensional. If $P$ is $n$-dimensional, then ${\mathcal P} (P) := {\mathcal P} (P, \, P)$ is the initial object in the category of all commutative bialgebras coacting on $P$. As an algebra, ${\mathcal P} (P)$ can be deescribed as the quotient of the polynomial algebra $k[X_{ij} \, | \, i, j = 1, \cdots, n]$ through an ideal generated by $2 n^3$ non-homogeneous polynomials of degree $\leq 2$. Two applications are provided. The first one describes the automorphisms group ${\rm Aut}_{\rm Poiss} (P)$ as the group of all invertible group-like elements of the finite dual ${\mathcal P} (P)^{\rm o}$. Secondly, we show that for an abelian group $G$, all $G$-gradings on $P$ can be explicitly described and classified in terms of the universal coacting bialgebra ${\mathcal P} (P)$., Comment: Continues arXiv:2006.00711, arXiv:2301.03051

A $4$-algebra is a commutative algebra $A$ over a field $k$ such that $(a^2)^2 = 0$, for all $a \in A$. We have proved recently \cite{Mil} that $4$-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let $A$ be a $4$-algebra, $E$ a vector space and $\pi : E \to A$ a surjective linear map with $V = {\rm Ker} (\pi)$. All $4$-algebra structures on $E$ such that $\pi : E \to A$ is an algebra map are described and classified by a global cohomological object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$. Any such $4$-algebra is isomorphic to a crossed product $V \# A$ and ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is a coproduct, over all $4$-algebras structures $\cdot_V$ on $V$, of all non-abelian cohomologies ${\mathbb H}^{2}_{\rm nab} \, \bigl(A, \, (V, \, \cdot_{V} )\bigl)$, which are the classifying objects for all extensions of $A$ by $V$. Several applications and examples are provided: in particular, ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ and ${\mathbb G} {\mathbb H}^{2} \, (k, \, V)$ are explicitly computed and the Galois group ${\rm Gal} \, (V \# A/ V )$ of the extension $V \hookrightarrow V \# A$ is described., Comment: The final version will appear in J. Algebra. arXiv admin note: text overlap with arXiv:1507.08146, arXiv:1503.05364

We prove that any Bernstein algebra $(A, \omega)$ is isomorphic to a semidirect product $V \ltimes_{(\cdot, \, \Omega)} \, k$ associated to a commutative algebra $(V, \cdot)$ such that $(x^2)^2 = 0$, for all $x\in A$ and an idempotent endomorphism $\Omega = \Omega^2 \in {\rm End}_k (V)$ of $V$ satisfying two compatibility conditions. The set of types of $(1 + |I|)$-dimensional Bernstein algebras is parametrized by an explicitely constructed (using linear algebra tools) classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups $(V, +) \ltimes {\rm GL}_k (V)$., Comment: The final version will appear in Journal of Algebra and its Applications

We investigate the factorization problem as well as the classifying complements problem in the setting of Jordan algebras. Matched pairs of Jordan algebras and the corresponding bicrossed products are introduced. It is shown that any Jordan algebra which factorizes through two given Jordan algebras is isomorphic to a bicrossed product associated to a certain matched pair between the same two Jordan algebras. Furthermore, a new type of deformation of a Jordan algebra is proposed as the main step towards solving the classifying complements problem., Comment: To appear in Collectanea Math.; Continues arXiv:1204.1805, arXiv:1205.6110, arXiv:1205.6564, arXiv:1311.6232, arXiv:1312.4018, arXiv:1603.01854, arXiv:1611.05674, arXiv:2107.04970

Given a Jordan algebra $A$ and a vector space $V$, we describe and classify all Jordan algebras containing $A$ as a subalgebra of codimension ${\rm dim}_k (V)$ in terms of a non-abelian cohomological type object ${\mathcal J}_{A} \, (V, \, A)$. Any such algebra is isomorphic to a newly introduced object called \emph{unified product} $A \, \natural \, V$. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology ${\rm H}^2_{\rm nab} \, (V, \, A )$ associated to two Jordan algebras $A$ and $V$ which classifies all extensions of $V$ by $A$ is also constructed. Several applications and examples are given: we prove that ${\rm H}^2_{\rm nab} \, (k, \, k^n)$ is identified with the set of all matrices $D\in M_n(k)$ satisfying $2\, D^3 - 3 \, D^2 + D = 0$., Comment: Continues arXiv:1011.1633, arXiv:1011.2174, arXiv:1301.5442, arXiv:1305.6022, arXiv:1307.2540, arXiv:1308.5559, arXiv:1309.1986, arXiv:1507.08146; arXiv:2105.14722; restates preliminaries and definitions for sake of clarity

For a given Jacobi-Jordan algebra $A$ and a vector space $V$ over a field $k$, a non-abelian cohomological type object ${\mathcal H}^{2}_{A} \, (V, \, A)$ is constructed: it classifies all Jacobi-Jordan algebras containing $A$ as a subalgebra of codimension equal to ${\rm dim}_k (V)$. Any such algebra is isomorphic to a so-called \emph{unified product} $A \, \natural \, V$. Furthermore, we introduce the bicrossed (semi-direct, crossed, or skew crossed) product $A \bowtie V$ associated to two Jacobi-Jordan algebras as a special case of the unified product. Several examples and applications are provided: the Galois group of the extension $A \subseteq A \bowtie V$ is described as a subgroup of the semidirect product of groups ${\rm GL}_k (V) \rtimes {\rm Hom}_k (V, \, A)$ and an Artin type theorem for Jacobi-Jordan algebra is proven. The key tools for classifying supersolvable and flag Jacobi-Jordan algebras are introduced., Comment: Continues arXiv:1011.1633, arXiv:1011.2174, arXiv:1301.5442, arXiv:1305.6022, arXiv:1307.2540, arXiv:1308.5559, arXiv:1309.1986, arXiv:1507.08146; restates preliminaries and definitions for sake of clarity. To appear in Linear Algebra and its Applications

Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $a (A)^{\rm o}$. For a group $G$, all $G$-gradings on $A$ are explicitly described and classified: the set of isomorphisms classes of all $G$-gradings on $A$ is in bijection with the quotient set $ {\rm Hom}_{\rm BiAlg} \, \bigl( a (A) , \, k[G] \bigl)/\approx$ of all bialgebra maps $a (A) \, \to k[G]$, via the equivalence relation implemented by the conjugation with an invertible group-like element of $a (A)^{\rm o}$., Comment: Continues (associative algebra version of) arXiv:2006.00711

For an $n$-dimensional Leibniz/Lie algebra $\mathfrak{h}$ over a field $k$ we introduce a new invariant ${\mathcal A}(\mathfrak{h})$, called the \emph{universal algebra} of $\mathfrak{h}$, as a quotient of the polynomial algebra $k[X_{ij} \, | \, i, j = 1, \cdots, n]$ through an ideal generated by $n^3$ polynomials. We prove that ${\mathcal A}(\mathfrak{h})$ admits a unique bialgebra structure which makes it an initial object among all commutative bialgebras coacting on $\mathfrak{h}$. The new object ${\mathcal A} (\mathfrak{h})$ is the key tool in answering two open problems in Lie algebra theory. First, we prove that the automorphism group ${\rm Aut}_{Lbz} (\mathfrak{h})$ of $\mathfrak{h}$ is isomorphic to the group $U \bigl( G({\mathcal A} (\mathfrak{h})^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual ${\mathcal A} (\mathfrak{h})^{\rm o}$. Secondly, for an abelian group $G$, we show that there exists a bijection between the set of all $G$-gradings on $\mathfrak{h}$ and the set of all bialgebra homomorphisms ${\mathcal A} (\mathfrak{h}) \to k[G]$. Based on this, all $G$-gradings on $\mathfrak{h}$ are explicitly classified and parameterized. ${\mathcal A} (\mathfrak{h})$ is also used to prove that there exists a universal commutative Hopf algebra associated to any finite dimensional Leibniz algebra $\mathfrak{h}$., Comment: Final version, to appear in Journal of Algebra

We introduce some basic concepts for Jacobi-Jordan algebras such as: representations, crossed products or Frobenius/metabelian/co-flag objects. A new family of solutions for the quantum Yang-Baxter equation is constructed arising from any $3$-step nilpotent Jacobi-Jordan algebra. Crossed products are used to construct the classifying object for the extension problem in its global form. For a given Jacobi-Jordan algebra $A$ and a given vector space $V$ of dimension $\mathfrak{c}$, a global non-abelian cohomological object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is constructed: it classifies, from the view point of the extension problem, all Jacobi-Jordan algebras that have a surjective algebra map on $A$ with kernel of dimension $\mathfrak{c}$. The object ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ responsible for the classification of co-flag algebras is computed, all $1 + {\rm dim} (A)$ dimensional Jacobi-Jordan algebras that have an algebra surjective map on $A$ are classified and the automorphism groups of these algebras is determined. Several examples involving special sets of matrices and symmetric bilinear forms as well as equivalence relations between them (generalizing the isometry relation) are provided., Comment: 24 pages; to appear in Linear Algebra and its Applications

If $\mathfrak{g} \subseteq \mathfrak{h}$ is an extension of Lie algebras over a field $k$ such that ${\rm dim}_k (\mathfrak{g}) = n$ and ${\rm dim}_k (\mathfrak{h}) = n + m$, then the Galois group ${\rm Gal} \, (\mathfrak{h}/\mathfrak{g})$ is explicitly described as a subgroup of the canonical semidirect product of groups ${\rm GL} (m, \, k) \rtimes {\rm M}_{n\times m} (k)$. An Artin type theorem for Lie algebras is proved: if a group $G$ whose order isinvertible in $k$ acts as automorphisms on a Lie algebra $\mathfrak{h}$, then $\mathfrak{h}$ is isomorphic to a skew crossed product $\mathfrak{h}^G \, \#^{\bullet} \, V$, where $\mathfrak{h}^G$ is the subalgebra of invariants and $V$ is the kernel of the Reynolds operator. The Galois group ${\rm Gal} \,(\mathfrak{h}/\mathfrak{h}^G)$ is also computed, highlighting the difference from the classical Galois theory of fields where the corresponding group is $G$. The counterpart for Lie algebras of Hilbert's Theorem 90 is proved and based on it the structure of Lie algebras $\mathfrak{h}$ having a certain type of action of a finite cyclic group is described. Radical extensions of finite dimensional Lie algebras are introduced and it is shown that their Galois group is solvable. Several applications and examples are provided., Comment: final version

Let $A$ be a unital associative algebra over a field $k$, $E$ a vector space and $\pi : E \to A$ a surjective linear map with $V = {\rm Ker} (\pi)$. All algebra structures on $E$ such that $\pi : E \to A$ becomes an algebra map are described and classified by an explicitly constructed global cohomological type object ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$. Any such algebra is isomorphic to a Hochschild product $A \star V$, an algebra introduced as a generalization of a classical construction. We prove that ${\mathbb G} {\mathbb H}^{2} \, (A, \, V)$ is the coproduct of all non-abelian cohomologies ${\mathbb H}^{2} \, \, (A, \, (V, \cdot))$. The key object ${\mathbb G} {\mathbb H}^{2} \, (A, \, k)$ responsible for the classification of all co-flag algebras is computed. All Hochschild products $A \star k$ are also classified and the automorphism groups ${\rm Aut}_{\rm Alg} (A \star k)$ are fully determined as subgroups of a semidirect product $A^* \, \ltimes \bigl(k^* \times {\rm Aut}_{\rm Alg} (A) \bigl)$ of groups. Several examples are given as well as applications to the theory of supersolvable coalgebras or Poisson algebras. In particular, for a given Poisson algebra $P$, all Poisson algebras having a Poisson algebra surjection on $P$ with a $1$-dimensional kernel are described and classified., Comment: Continues arXiv:1308.5559, arXiv:1309.1986 and arXiv:1305.6022; restates preliminaries and definitions for sake of clarity. Final version to appear in J. Pure Appl. Algebra

Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra $A$ and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vector space $V$ a non-abelian cohomological type object ${\mathcal J}{\mathcal H}^{2} \, (V, \, A)$ is constructed: it classifies all Jacobi algebras containing $A$ as a subalgebra of codimension equal to ${\rm dim} (V)$. Representations of $A$ are used in order to give the decomposition of ${\mathcal J}{\mathcal H}^{2} \, (V, \, A)$ as a coproduct over all Jacobi $A$-module structures on $V$. The bicrossed product $P \bowtie Q$ of two Poisson algebras recently introduced by Ni and Bai appears as a special case of our construction. A new type of deformations of a given Poisson algebra $Q$ is introduced and a cohomological type object $\mathcal{H}\mathcal{A}^{2} \bigl(P,\, Q ~|~ (\triangleleft, \, \triangleright, \, \leftharpoonup, \, \rightharpoonup)\bigl)$ is explicitly constructed as a classifying set for the bicrossed descent problem for extensions of Poisson algebras. Several examples and applications are provided., Comment: 40 pages; to appear in Journal of Noncommutative Geometry

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 Algebra

Metabelian algebras are introduced and it is shown that an algebra $A$ is metabelian if and only if $A$ is a nilpotent algebra having the index of nilpotency at most $3$, i.e. $x y z t = 0$, for all $x$, $y$, $z$, $t \in A$. We prove that the It\^{o}'s theorem for groups remains valid for associative algebras. A structure theorem for metabelian algebras is given in terms of pure linear algebra tools and their classification from the view point of the extension problem is proven. Two border-line cases are worked out in detail: all metabelian algebras having the derived algebra of dimension $1$ (resp. codimension $1$) are explicitly described and classified. The algebras of the first family are parameterized by bilinear forms and classified by their homothetic relation. The algebras of the second family are parameterized by the set of all matrices $(X, Y, u) \in {\rm M}_{n}(k)^2 \times k^n$ satisfying $X^2 = Y^2 = 0$, $XY = YX$ and $Xu = Yu$., Comment: the final version will appear in Bulletin of the Malaysian Mathematical Sciences Society

Let $P$ be a Poisson algebra, $E$ a vector space and $\pi : E \to P$ an epimorphism of vector spaces with $V = {\rm Ker} (\pi)$. The global extension problem asks for the classification of all Poisson algebra structures that can be defined on $E$ such that $\pi : E \to P$ becomes a morphism of Poisson algebras. From a geometrical point of view it means to decompose this groupoid into connected components and to indicate a point in each such component. All such Poisson algebra structures on $E$ are classified by an explicitly constructed classifying set ${\mathcal G} {\mathcal P} {\mathcal H}^{2} \, (P, \, V)$ which is the coproduct of all non-abelian cohomological objects ${\mathcal P} {\mathcal H}^{2} \, (P, \, (V, \cdot_V, [-,-]_V))$ which are the classifying sets for all extensions of $P$ by $(V, \cdot_V, [-,-]_V)$. The second classical Poisson cohomology group $H^2 (P, V)$ appears as the most elementary piece among all components of ${\mathcal G} {\mathcal P} {\mathcal H}^{2} \, (P, \, V)$. Several examples are provided in the case of metabelian Poisson algebras or co-flag Poisson algebras over $P$: the latter being Poisson algebras $Q$ which admit a finite chain of epimorphisms of Poisson algebras $P_n : = Q \stackrel{\pi_{n}}{\longrightarrow} P_{n-1} \, \cdots \, P_1 \stackrel{\pi_{1}} {\longrightarrow} P_{0} := P$ such that ${\rm dim} ( {\rm Ker} (\pi_{i}) ) = 1$, for all $i = 1, \cdots, n$., Comment: Final version; to appear in J. Algebra. Continues arXiv:1301.5442, arXiv:1305.6022, arXiv:1307.2540, arXiv:1308.5559; restates preliminaries and definitions for sake of clarity

Let $\mathfrak{g}$ be a Leibniz algebra and $E$ a vector space containing $\mathfrak{g}$ as a subspace. All Leibniz algebra structures on $E$ containing $\mathfrak{g}$ as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: ${\mathcal H}{\mathcal L}^{2}_{\mathfrak{g}} \, (V, \, \mathfrak{g})$ provides the classification up to an isomorphism that stabilizes $\mathfrak{g}$ and ${\mathcal H}{\mathcal L}^{2} \, (V, \, \mathfrak{g})$ will classify all such structures from the view point of the extension problem - here $V$ is a complement of $\mathfrak{g}$ in $E$. A general product, called the unified product, is introduced as a tool for our approach. The crossed (resp. bicrossed) products between two Leibniz algebras are introduced as special cases of the unified product: the first one is responsible for the extension problem while the bicrossed product is responsible for the factorization problem. The description and the classification of all complements of a given extension $\mathfrak{g} \subseteq \mathfrak{E} $ of Leibniz algebras are given as a converse of the factorization problem. They are classified by another cohomological object denoted by ${\mathcal H}{\mathcal A}^{2}(\mathfrak{h}, \mathfrak{g} \, | \, (\triangleright, \triangleleft, \leftharpoonup, \rightharpoonup))$, where $(\triangleright, \triangleleft, \leftharpoonup, \rightharpoonup)$ is the canonical matched pair associated to a given complement $\mathfrak{h}$. Several examples are worked out in details., Comment: To appear in Linear Algebra and its Applications. Continues arXiv:1301.5442, arXiv:1305.6022; restates preliminaries and definitions for sake of clarity

Let $A$ be a unital associative algebra over a field $k$. All unital associative algebras containing $A$ as a subalgebra of a given codimension $\mathfrak{c}$ are described and classified. For a fixed vector space $V$ of dimension $\mathfrak{c}$, two non-abelian cohomological type objects are explicitly constructed: ${\mathcal A}{\mathcal H}^{2}_{A} \, (V, \, A)$ will classify all such algebras up to an isomorphism that stabilizes $A$ while ${\mathcal A}{\mathcal H}^{2} \, (V, \, A)$ provides the classification from H\"{o}lder's extension problem viewpoint. A new product, called the unified product, is introduced as a tool of our approach. The classical crossed product or the twisted tensor product of algebras are special cases of the unified product. Two main applications are given: the Galois group ${\rm Gal} \, (B/A)$ of an extension $A \subseteq B$ of associative algebras is explicitly described as a subgroup of a semidirect product of groups ${\rm GL}_k (V) \rtimes {\rm Hom}_k (V, \, A)$, where the vector space $V$ is a complement of $A$ in $B$. The second application refers to supersolvable algebras introduced as the associative algebra counterpart of supersolvable Lie algebras. Several explicit examples are given for supersolvable algebras over an arbitrary base field, including those of characteristic two whose difficulty is illustrated., Comment: 30 pages; new version: title changed; added a new section on Galois extensions of associative algebras. Final version to appear in Monatsh. fur Mathematik. DOI:10.1007/s00605-015-0814-8

Let $\mathfrak{g}$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g}$ as a subspace. The paper is devoted to the \emph{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 ofthe 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., Comment: To appear in Monatshefte f\"ur Mathematik

Let $A$ be an algebra over a commutative ring $k$. It is known that the categories of non-commutative descent data, of comodules over the Sweedler canonical coring, of right $A$-modules with a flat connection are isomorphic as braided monoidal categories to the center of the category of $A$-bimodules. We prove that the braiding on these categories is unique if there exists a $k$-linear unitary map $E : A \to Z(A)$. This condition is satisfied if $k$ is a field or $A$ is a commutative or a separable algebra., Comment: 9 pages, submitted

For a given Hopf algebra $A$ we classify all Hopf algebras $E$ that are coalgebra split extensions of $A$ by $H_4$, where $H_4$ is the Sweedler's 4-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $A # H_4$ by computing explicitly two classifying objects: the cohomological 'group' ${\mathcal H}^{2} (H_4, A)$ and $\textsc{C}\textsc{r}\textsc{p} (H_4, A) :=$ the set of types of isomorphisms of all crossed products $A # H_4$. All crossed products $A #H_4$ are described by generators and relations and classified: they are parameterized by the set ${\mathcal Z}{\mathcal P} (A)$ of all central primitive elements of $A$. Several examples are worked out in detail: in particular, over a field of characteristic $p \geq 3$ an infinite family of non-isomorphic Hopf algebras of dimension $4p$ is constructed. The groups of automorphisms of these Hopf algebras are also described., Comment: 21 pages; substantial changes from previous version; to appear in J. Algebra Appl

Let $A \subseteq E$ be a given extension of Hopf (respectively Lie) algebras. We answer the \emph{classifying complements problem} (CCP) which consists of describing and classifying all complements of $A$ in $E$. If $H$ is a given complement then all the other complements are obtained from $H$ by a certain type of deformation. We establish a bijective correspondence between the isomorphism classes of all complements of $A$ in $E$ and a cohomological type object ${\mathcal H}{\mathcal A}^{2} (H, A \, | \, (\triangleright, \triangleleft) )$, where $(\triangleright, \triangleleft)$ is the matched pair associated to $H$. The factorization index $[E: A]^f$ is introduced as a numerical measure of the (CCP). For two $n$-th roots of unity we construct a $4n^2$-dimensional Hopf algebra whose factorization index over the group algebra is arbitrary large., Comment: 18 pages, to appear in Journal of Algebra

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 $A \bowtie H$ 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 $CoZ^{1} (H, A) \times \Aut_{\rm CoAlg}^1 (H)$, where $CoZ^{1} (H, A)$ is the group of unitary cocentral maps and $\Aut_{\rm CoAlg}^1(H)$ is the group of unitary automorphisms of the coalgebra $H$. Among several applications and examples, all bicrossed products $H_4 \bowtie k[C_n]$ are described by generators and relations and classified: they are quantum groups at roots of unity $H_{4n, \omega}$ which are classified by pure arithmetic properties of the ring $\mathbb{Z}_n$. 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 $\Aut_{\rm Hopf}(H_{4n, \omega})$ of Hopf algebra automorphisms is fully described., Comment: 36 pages; corrected minor typos; to appear in Algebras and Representation Theory

Let $A \leq G$ be a subgroup of a group $G$. An $A$-complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A \cap H = \{1\}$. The \emph{classifying complements problem} asks for the description and classification of all $A$-complements of $G$. We shall give the answer to this problem in three steps. Let $H$ be a given $A$-complement of $G$ and $(\triangleright, \triangleleft)$ the canonical left/right actions associated to the factorization $G = A H$. To start with, $H$ is deformed to a new $A$-complement of $G$, denoted by $H_r$, using a certain map $r: H \to A$ called a deformation map of the matched pair $(A, H, \triangleright, \triangleleft)$. Then the description of all complements is given: ${\mathbb H}$ is an $A$-complement of $G$ if and only if ${\mathbb H}$ is isomorphic to $H_{r}$, for some deformation map $r: H \to A$. Finally, the classification of complements proves that there exists a bijection between the isomorphism classes of all $A$-complements of $G$ and a cohomological object ${\mathcal D} \, (H, A \, | \,(\triangleright, \triangleleft))$. As an application we show that the theoretical formula for computing the number of isomorphism types of all groups of order $n$ arises only from the factorization $S_n = S_{n-1} C_n$., Comment: 13 pages; to appear in Ann. Inst. Fourier

Let $A$ be an algebra over a commutative ring $k$. We compute the center of the category of $A$-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical $A$-coring, Yetter-Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical $A$-coring $A \ot A$ is braided monoidal. We provide several applications: for instance, if $A$ is finitely generated projective over $k$ then the category of left End_k(A)$-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of $A$. As another application, new families of solutions for the quantum Yang-Baxter equation are constructed: they are canonical maps $\Omega$ associated to any right comodule over the Sweedler canonical coring $A \ot A$ and satisfy the condition $\Omega^3 = \Omega$. Explicit examples are provided., Comment: 14 pages; to appear in J. Algebra Appl

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 $A\ot A\ot A$ satisfying certain axioms. We show that all braidings are symmetries. If $A$ is commutative, then there exists a braiding on ${}_A\Mm_A$ if and only if $k\to A$ is an epimorphism in the category of rings, and then the corresponding $R$-matrix is trivial. If the invariants functor $G = (-)^A:\{}_A\Mm_A\to \Mm_k$ 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., Comment: 12 pages; final version, to appear in Appl. Cat. Structures

The unified product was defined in \cite{am3} related to the restricted extending structure problem for Hopf algebras: a Hopf algebra $E$ factorizes through a Hopf subalgebra $A$ and a subcoalgebra $H$ such that $1\in H$ if and only if $E$ is isomorphic to a unified product $A \ltimes H$. Using the concept of normality of a morphism of coalgebras in the sense of Andruskiewitsch and Devoto we prove an equivalent description for the unified product from the point of view of split morphisms of Hopf algebras. A Hopf algebra $E$ is isomorphic to a unified product $A \ltimes H$ if and only if there exists a morphism of Hopf algebras $i: A \rightarrow E$ which has a retraction $\pi: E \to A$ that is a normal left $A$-module coalgebra morphism. A necessary and sufficient condition for the canonical morphism $i : A \to A\ltimes H$ to be a split monomorphism of bialgebras is proved, i.e. a condition for the unified product $A\ltimes H$ to be isomorphic to a Radford biproduct $L \ast A$, for some bialgebra $L$ in the category $_{A}^{A}{\mathcal YD}$ of Yetter-Drinfel'd modules. As a consequence, we present a general method to construct unified products arising from an unitary not necessarily associative bialgebra $H$ that is a right $A$-module coalgebra and a unitary coalgebra map $\gamma : H \to A$ satisfying four compatibility conditions. Such an example is worked out in detail for a group $G$, a pointed right $G$-set $(X, \cdot, \lhd)$ and a map $\gamma : G \to X$., Comment: 16 pages, to appear in AMS Contemporary Math

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., Comment: 24 pages, 3 figures. Final version, to appear in Journal of Algebra

Let $H$ be a group and $E$ a set such that $H \subseteq E$. We shall describe and classify up to an isomorphism of groups that stabilizes $H$ the set of all group structures that can be defined on $E$ such that $H$ is a subgroup of $E$. A general product, which we call the unified product, is constructed such that both the crossed product and the bicrossed product of two groups are special cases of it. It is associated to $H$ and to a system $\bigl((S, 1_S,\ast), \triangleleft, \, \triangleright, \, f \bigl)$ called a group extending structure and we denote it by $H \ltimes S$. There exists a group structure on $E$ containing $H$ as a subgroup if and only if there exists an isomorphism of groups $(E, \cdot) \cong H \ltimes S$, for some group extending structure $\bigl((S, 1_S,\ast), \triangleleft, \, \triangleright, \, f \bigl)$. All such group structures on $E$ are classified up to an isomorphism of groups that stabilizes $H$ by a cohomological type set ${\mathcal K}^{2}_{\ltimes} (H, (S, 1_S))$. A Schreier type theorem is proved and an explicit example is given: it classifies up to an isomorphism that stabilizes $H$ all groups that contain $H$ as a subgroup of index 2., Comment: 17 pages; to appear in Algebras and Representation Theory

We call a monoidal category ${\mathcal C}$ a Serre category if for any $C$, $D \in {\mathcal C}$ such that $C\ot D$ is semisimple, $C$ and $D$ are semisimple objects in ${\mathcal C}$. Let $H$ be an involutory Hopf algebra, $M$, $N$ two $H$-(co)modules such that $M \otimes N$ is (co)semisimple as a $H$-(co)module. If $N$ (resp. $M$) is a finitely generated projective $k$-module with invertible Hattory-Stallings rank in $k$ then $M$ (resp. $N$) is (co)semisimple as a $H$-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel'd modules over $H$ the dimension of which is invertible in $k$ are Serre categories., Comment: a new version: 8 pages

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, \alpha, \beta)$ is deformed using a combinatorial datum $(\sigma, v, r)$ consisting of an automorphism $\sigma$ 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 $\bigl(H, (G,*), \alpha', \beta' \bigl)$ such that there exist an $\sigma$-invariant isomorphism of groups $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} (G,*)$. Moreover, if we fix the group $H$ and the automorphism $\sigma \in \Aut(H)$ then any $\sigma$-invariant isomorphism $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} G'$ 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., Comment: 21 pages, final version to appear in Central European J. Math

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., Comment: Final version. To appear in AJSE

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., Comment: Final version: to appear in Algebras and Representation Theory

We introduce and discuss the notion of naturally full functor. The definition is similar to the definition of separable functor: a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful functor. We study general properties of naturally full functors. We also discuss when functors between module categories and between categories of comodules over a coring are naturally full., Comment: 27 pages

We introduce separable functors of the second kind (or $H$-separable functors) and $H$-Maschke functors. $H$-separable functors are generalizations of separable functors. Various necessary and sufficient conditions for a functor to be $H$-separable or $H$-Maschke, in terms of generalized (co)Casimir elements (integrals, in the case of Hopf algebras), are given. An $H$-separable functor is always $H$-Maschke, but the converse holds in particular situations. A special role will be played by Frobenius functors and their relations to $H$-separability. Our concepts are applied to modules, comodules, entwined modules, quantum Yetter-Drinfeld modules, relative Hopf modules., Comment: 23 pages

A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second type if $G$ is a left adjoint of $\beta F\alpha$ for some category equivalences $\alpha$ and $\beta$. Frobenius ring extensions of the second kind provide examples of Frobenius pairs of the second kind. We study Frobenius pairs of the second kind between categories of modules, comodules, and comodules over a coring. We also show that a finitely generated projective Hopf algebra over a commutative ring is always a Frobenius extension of the second kind, and prove that the integral spaces of the Hopf algebra and its dual are isomorphic., Comment: 23 pages; we have a corrected a mistake in Theorem 3.4 and Corollary 3.5

We introduce and study a general concept of integral of a threetuple (H, A, C), where H is a Hopf algebra acting on a coalgebra C and coacting on an algebra A. In particular, quantum integrals associated to Yetter-Drinfel'd modules are defined. Let A be an H-bicomodule algebra, $^H {\cal YD}_A$ be the category of (generalized) Yetter-Drinfel'd modules and $B$ the subalgebra of coinvariants of the Verma structure of $A$. We introduce the concept of quantum Galois extensions and we prove the affineness criterion in a quantum version., Comment: latex 32 pg. J. Algebra, to appear

The study of the pentagon (fusion) equation leds to the Structure and the Classification theorem for finite dimenasional Hopf algebras: there exists a one to one correspondence between the set of types of n-dimensional Hopf algebtras and the set of the orbits of the resticted Jordan action $GL_n(k) \times M_n(k)\otimes M_n(k) \to M_n(k) \otimes M_nk$ $(u, R) \to (u\otimes u)R (u\otimes u)^{-1}$, the representatives of wich are invertible solutions of length n of the pentagon equation., Comment: 22 pg, latex

We study when induction functors (and their adjoints) between categories of Doi-Hopf modules and, more generally, entwined modules are separable, resp. Frobenius. We present a unified approach, leading to new proofs of old results by the authors, as well as to some new ones. Also our methods provide a categorical explanation for the relationship between separability and Frobenius properties., Comment: 28 pages, LaTeX; Contribution to the proceedings of the ``Fifth week on algebra and geometry (SAGA5)'', Leon, Spain 1999

We present a unified apoach to the study of separable and Frobenius algebras. The crucial observation is thsat both cases are related to the nonlinear equation $R^{12}R^{23}=R^{23}R^{13}=R^{13}R^{12}$, called the FS-equation. Given a solution of the FS-equation satisfying certain normalizing conditions, we can construct a Frobenius algebra or a separable algebra A(R). The main result of this paper in the structure of this two fundamental types of algebras: a finitely generated projective Frobenius or separable $k$-algebra $A$ is isomorphic to such A(R). If $A$ is free as a $k$-module, then A(R) can be described using generators and relations. A new characterisation of Frobenius extensions is given: $B\subset A$ is Frobenius if and only if $A$ has a $B$-coring structure such that the comultiplication $\Delta$ is an $A$-bimodule map., Comment: 34 pages