arXiv subjects

G. Militaru

Publications and source records attributed to G. Militaru.

At least 37 records · Page 2Linked to original sources

arXiv2014

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 $S_n = S_{n-1} C_n$.

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arXiv2013

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.

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arXiv2013

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.

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arXiv2013

Classifying complements for Hopf algebras and Lie algebras

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.

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arXiv2013

Extending structures I: the level of groups

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.

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arXiv2012

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 $A \bowtie H$ 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 $H_{4n, ω}$ 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 $\Aut_{\rm Hopf}(H_{4n, ω})$ of Hopf algebra automorphisms is fully described.

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arXiv2012

The uniqueness of braidings on the monoidal category of non-commutative descent data

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.

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arXiv2012

Classifying coalgebra split extensions of Hopf algebras

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.

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arXiv2012

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 $A\ot A\ot A$ 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.

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arXiv2012

Unified products and split extensions of Hopf algebras

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 $π: E \to A$ 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 $γ: H \to A$ 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 $γ: G \to X$.

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arXiv2011

The center of the category of bimodules and descent data for non-commutative rings

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 $Ω$ associated to any right comodule over the Sweedler canonical coring $A \ot A$ and satisfy the condition $Ω^3 = Ω$. Explicit examples are provided.

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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.

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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.

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arXiv2009

Serre Theorem for involutory Hopf algebras

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.

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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.

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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.

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arXiv2005

Frobenius Functors of the second kind

A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second type if $G$ is a left adjoint of $βFα$ for some category equivalences $α$ and $β$. 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.

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arXiv2004

Naturally full functors in nature

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.

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