arXiv subjects

Gigel Militaru

Publications and source records attributed to Gigel Militaru.

12 recordsLinked to original sources

arXiv2026

Centralizers of maps as automorphism groups of Yang--Baxter solutions

For a partition $X = Y \sqcup Z$ of a set $X$ and an arbitrary map $g \colon Y \to {\rm Hom}_{\rm Set} (Z, \, Z)$, we construct a solution $R_g \colon X \times X \to X \times X$ of the braid equation. We characterize intrinsically the solutions which are isomorphic to some $R_g$ and we give a necessary and sufficient condition for two solutions $R_g$ and $R_{g'}$ to be isomorphic. Consequently, the number of isomorphism classes of the solutions $R_g$ on a set with $n$ elements is $1 + \sum_{k = 1}^{n-1} \widetilde{o} (n - k, \, k)$, where $\widetilde{o} (m, \, k)$ is the number of orbits of the symmetric group $S_m$, acting by conjugation, on the set of multisets with $k$ elements of maps $\{1, \ldots, m\} \to \{1, \ldots, m\}$. This construction is then used to obtain a solution of the braid equation whose automorphism group is exactly the centralizer ${\rm Aut} (f_1, \ldots, f_k)$ of a given family $f_1, \ldots, f_k \colon X \to X$ of maps, $k \geq 1$. For $k = 2$, combined with a result of Agore, Chirvasitu and the author, this gives another proof of the fact that any group is isomorphic to the automorphism group of a solution of the braid equation (equivalently, of the set-theoretic Yang--Baxter equation).

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arXiv2026

Bipartite solutions of the set-theoretic Yang--Baxer equation

We introduce a new class of unitary set-theoretic Yang--Baxter solutions, called \emph{bipartite} solutions. This class, which is purely combinatorial in nature, forms a full subcategory of $\mathcal{LYB}$, the category of Lyubashenko type solutions of Agore, Chirvasitu and the author, and is particularly amenable to explicit classification results. We give an intrinsic necessary and sufficient condition for a solution of the Yang--Baxter equation to be isomorphic to a bipartite solution, we settle the isomorphism problem within this class and, as an application, we determine their automorphism groups. If $X$ is a finite set with $|X| = n$, the number of isomorphism classes of bipartite solutions on $X$ is $\lfloor n/2 \rfloor + 1 + \sum_{m=2}^{n} \bigl( o(m, n-m) - 1 \bigr)$, where $o(m, k)$ denotes the number of orbits of the symmetric group $S_m$ acting by conjugation on the $k$-element multisets of elements of $S_m$. The paper also offers a comparative study: for small $|X| = n$ we count, up to isomorphism, all set-theoretic Yang--Baxter solutions on $X$, together with five distinguished families of solutions. Throughout the range we were able to compute, the class $\mathcal{LYB}$ accounts for a large share of all bijective and of all non-degenerate solutions, strikingly close to the total.

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arXiv2014

Bicrossed Products, Matched Pair Deformations and the Factorization Index for Lie Algebras

For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product $\mathfrak{h} \ltimes (k^* \times {\rm Aut}_{\rm Lie} (\mathfrak{h}))$. In the non-perfect case the classification of these Lie algebras is a difficult task. Let $\mathfrak{l} (2n+1, k)$ be the Lie algebra with the bracket $[E_i, G] = E_i$, $[G, F_i] = F_i$, for all $i = 1, \dots, n$. We explicitly describe all Lie algebras containing $\mathfrak{l} (2n+1, k)$ as a subalgebra of codimension $1$ by computing all possible bicrossed products $k \bowtie \mathfrak{l} (2n+1, k)$. They are parameterized by a set of matrices ${\rm M}_n (k)^4 \times k^{2n+2}$ which are explicitly determined. Several matched pair deformations of $\mathfrak{l} (2n+1, k)$ are described in order to compute the factorization index of some extensions of the type $k \subset k \bowtie \mathfrak{l} (2n+1, k)$. We provide an example of such extension having an infinite factorization index.

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arXiv2014

The Classification of All Crossed Products $H_4 \# k[C_{n}]$

Using the computational approach introduced in [Agore A.L., Bontea C.G., Militaru G., J. Algebra Appl. 12 (2013), 1250227, 24 pages, arXiv:1207.0411] we classify all coalgebra split extensions of $H_4$ by $k[C_n]$, where $C_n$ is the cyclic group of order $n$ and $H_4$ is Sweedler's $4$-dimensional Hopf algebra. Equivalently, we classify all crossed products of Hopf algebras $H_4 \# k[C_{n}]$ by explicitly computing two classifying objects: the cohomological 'group' ${\mathcal H}^{2} ( k[C_{n}], H_4)$ and $\text{CRP}( k[C_{n}], H_4):=$ the set of types of isomorphisms of all crossed products $H_4 \# k[C_{n}]$. More precisely, all crossed products $H_4 \# k[C_n]$ are described by generators and relations and classified: they are $4n$-dimensional quantum groups $H_{4n, λ, t}$, parameterized by the set of all pairs $(λ, t)$ consisting of an arbitrary unitary map $t : C_n \to C_2$ and an $n$-th root $λ$ of $\pm 1$. As an application, the group of Hopf algebra automorphisms of $H_{4n, λ, t}$ is explicitly described.

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arXiv2013

The global extension problem, co-flag and metabelian Leibniz algebras

Let $\mathfrak{L}$ be a Leibniz algebra, $E$ a vector space and $π: E \to \mathfrak{L}$ an epimorphism of vector spaces with $ \mathfrak{g} = {\rm Ker} (π)$. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined on $E$ such that $π: E \to \mathfrak{L}$ is a morphism of Leibniz algebras: from a geometrical viewpoint this means to give the decomposition of the groupoid of all such structures in its connected components and to indicate a point in each component. All such Leibniz algebra structures on $E$ are classified by a global cohomological object ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ which is explicitly constructed. It is shown that ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$ is the coproduct of all local cohomological objects $ {\mathbb H} {\mathbb L}^{2} \, \, (\mathfrak{L}, \, (\mathfrak{g}, [-,-]_{\mathfrak{g}}))$ that are classifying sets for all extensions of $\mathfrak{L}$ by all Leibniz algebra structures $(\mathfrak{g}, [-,-]_{\mathfrak{g}})$ on $\mathfrak{g}$. The second cohomology group ${\rm HL}^2 \, (\mathfrak{L}, \, \mathfrak{g})$ of Loday and Pirashvili appears as the most elementary piece among all components of ${\mathbb G} {\mathbb H} {\mathbb L}^{2} \, (\mathfrak{L}, \, \mathfrak{g})$. Several examples are worked out in details for co-flag Leibniz algebras over $\mathfrak{L}$, i.e. Leibniz algebras $\mathfrak{h}$ that have a finite chain of epimorphisms of Leibniz algebras $\mathfrak{L}_n : = \mathfrak{h} \stackrel{π_{n}}{\longrightarrow} \mathfrak{L}_{n-1} \, \cdots \, \mathfrak{L}_1 \stackrel{π_{1}} {\longrightarrow} \mathfrak{L}_{0} := \mathfrak{L}$ such that ${\rm dim} ({\rm Ker} (π_{i})) = 1$, for all $i = 1, \cdots, n$.

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arXiv2010

Four problems regarding representable functors

Let $R$, $S$ be two rings, $C$ an $R$-coring and ${}_{R}^C{\mathcal M}$ the category of left $C$-comodules. The category ${\bf Rep}\, ( {}_{R}^C{\mathcal M}, {}_{S}{\mathcal M} )$ of all representable functors ${}_{R}^C{\mathcal M} \to {}_{S}{\mathcal M}$ is shown to be equivalent to the opposite of the category ${}_{R}^C{\mathcal M}_S$. For $U$ an $(S,R)$-bimodule we give necessary and sufficient conditions for the induction functor $U\otimes_R - : {}_{R}^C\mathcal{M} \to {}_{S}\mathcal{M}$ to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings.

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arXiv2001

Bialgebroids, $\times_{A}$-bialgebras and duality

An equivalence between Lu's bialgebroids, Xu's bialgebroids with an anchor and Takeuchi's $\times_{A}$-bialgebras is explicitly proven. A new class of examples of bialgebroids is constructed. A (formal) dual of a bialgebroid, termed bicoalgebroid, is defined. A weak Hopf algebra is shown to be an example of such a bicoalgebroid.

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arXiv2001

Doi-Koppinen modules for quantum groupoids

A definition of a Doi-Koppinen datum over a noncommutative algebra is proposed. The idea is to replace a bialgebra in a standard Doi-Koppinen datum with a bialgebroid. The corresponding category of Doi-Koppinen modules over a noncommutative algebra is introduced. A weak Doi-Koppinen datum and module of [G. Bohm. Comm. Algebra, 28:4687--4698, 2000] are shown to be examples of a Doi-Koppinen datum and module over an algebra. A coring associated to a Doi-Koppinen datum over an algebra is constructed and various properties of induction and forgetful functors for Doi-Koppinen modules over an algebra are deduced from the properies of corresponding functors in the category of comodules of a coring.

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arXiv1998

A class of non-symmetric solutions for the integrability condition of the Knizhnik-Zamolodchikov equation: a Hopf algebra approach

This paper studied what shall be called the Long equation: that is the system of nonlinear equation $R^{12}R^{13}=R^{13}R^{12}$ and $R^{12}R^{23}= R^{23}R^{12}$. Any solution of this system supplies us a solution for the integrability solution of Knizhnic-Zamolodchikov equation. We shall approach this equation by introducing a new class of bialgebras, which we call Long bialgebras. A FRT type theorem is given: in the finite case any solution of the above equation is a co-homotety where the vector spase is a comodule over a Long bialgebra.

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arXiv1998

The Long dimodule category and nonlinear equation

We study the Long dimodule category in conection with a nonlinear equation; we called the D-equation. The category of Long dimodules will play for the D-equation the same role as the category of Yetter-Drinfel'd (crossed) modules play for the quantum Yang-Baxter equation. A FRT type theorem is given and new class of coalgebras are introduced. Several new examples of noncommutative noncocomutative bialgebras are given.

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arXiv1998

The Hopf modules category and the Hopf equation

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules category. As an application, a five dimensional noncommutative noncocommutative bialgebra is given.

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arXiv1998

New types of bialgebras arising from the Hopf equation

New types of bialgebras arising from the Hopf equation (pentagonal equation) are introduced and studied. They will play from the Hopf equation the same role as the co-quasitriangular do from the quantum Yang Baxter equation.

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