Extending structures, Galois groups and supersolvable associative algebras

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-

Hochschild products and global non-abelian cohomology for algebras. Applications

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. 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

Galois groups and group actions on Lie algebras

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 versio

On a type of commutative algebras

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 Application

Jacobi and Poisson algebras

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 Geometr

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 \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 Mathemati

Classifying complements for groups. Applications

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. Fourie

The global extension problem, crossed products and co-flag non-commutative Poisson algebras

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 clarit

Unified products for Leibniz algebras. Applications

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 clarit