arXiv subjects

A. L. Agore

Publications and source records attributed to A. L. Agore.

At least 19 recordsLinked to original sources

arXiv2026

Płonka bi-magmas and the set-theoretic Yang--Baxter equation

We introduce a new variety of non-associative algebras, called \emph{Płonka bi-magmas}, and use them to construct novel solutions to the set-theoretic Yang-Baxter (YB) equation. We establish explicit structural and classification results for Płonka bi-magmas, which allow for a detailed study of the induced families of YB-solutions. In particular, we identify and classify several naturally arising classes of YB-solutions from the universal algebra perspective, including the so-called bi-connected and ideal-simple ones. For instance, we prove that there are only countably many isomorphism classes of induced YB-solutions which are ideal-simple, and characterize them in terms of the odometer transformations familiar from ergodic theory. In particular, if $X$ is a finite set with $|X| = n$, then the number of isomorphism classes of all ideal-simple YB-solutions on $X$ is equal to the sum of all divisors of $n$.

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arXiv2025

The set-theoretic Yang-Baxter equation, Kimura semigroups and functional graphs

We prove 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) $τ(n) + \sum_{d|n}\left\lfloor \frac d2\right\rfloor$, where $τ(n)$ 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 $\mathrm{Aut}(f)$.

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arXiv2024

Functors between representation categories. Universal modules

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras with $\mathfrak{h}$ finite dimensional and consider ${\mathcal A} = {\mathcal A} (\mathfrak{h}, \, \mathfrak{g})$ to be the corresponding universal algebra as introduced in \cite{am20}. Given an ${\mathcal A}$-module $U$ and a Lie $\mathfrak{h}$-module $V$ we show that $U \otimes V$ can be naturally endowed with a Lie $\mathfrak{g}$-module structure. This gives rise to a functor between the category of Lie $\mathfrak{h}$-modules and the category of Lie $\mathfrak{g}$-modules and, respectively, to a functor between the category of ${\mathcal A}$-modules and the category of Lie $\mathfrak{g}$-modules. Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal ${\mathcal A}$-modules and universal Lie $\mathfrak{h}$-modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects \cite{Manin, Tambara}.

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arXiv2023

Universal constructions for Poisson algebras. Applications

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 ${\mathcal P} (P)$.

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arXiv2022

The factorization problem for Jordan algebras. 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.

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arXiv2022

Unified products for Jordan algebras. Applications

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 $2\, D^3 - 3 \, D^2 + D = 0$.

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arXiv2021

Algebraic constructions for Jacobi-Jordan algebras

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.

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arXiv2020

A new invariant for finite dimensional Leibniz/Lie algebras

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 $\mathfrak{h}$.

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arXiv2020

Universal coacting Poisson Hopf algebras

We introduce the analogue of Manin's universal coacting (bialgebra) Hopf algebra for Poisson algebras. First, for two given Poisson algebras $P$ and $U$, where $U$ is finite dimensional, we construct a Poisson algebra $\mathcal{B}(P,\, U)$ together with a Poisson algebra homomorphism $ψ_{\mathcal{B}(P,\,U)} \colon P \to U \otimes \mathcal{B}(P,\, U)$ satisfying a suitable universal property. $\mathcal{B}(P,\, U)$ is shown to admit a Poisson bialgebra structure for any pair of Poisson algebra homomorphisms subject to certain compatibility conditions. If $P=U$ is a finite dimensional Poisson algebra then $\mathcal{B}(P) = \mathcal{B}(P,\, P)$ admits a unique Poisson bialgebra structure such that $ψ_{\mathcal{B}(P)}$ becomes a Poisson comodule algebra and, moreover, the pair $\bigl(\mathcal{B}(P),\, ψ_{\mathcal{B}(P)}\bigl)$ is the universal coacting bialgebra of $P$. The universal coacting Poisson Hopf algebra $\mathcal{H}(P)$ on $P$ is constructed as the initial object in the category of Poisson comodule algebra structures on $P$ by using the free Poisson Hopf algebra on a Poisson bialgebra (\cite{A1}).

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arXiv2019

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.

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arXiv2019

Bicrossed products with the Taft algebra

Let $G$ be a group which admits a generating set consisting of finite order elements. We prove that any Hopf algebra which factorizes through the Taft algebra and the group Hopf algebra $K[G]$ (equivalently, any bicrossed product between the aforementioned Hopf algebras) is isomorphic to a smash product between the same two Hopf algebras. The classification of these smash products is shown to be strongly linked to the problem of describing the group automorphisms of $G$. As an application, we completely describe by generators and relations and classify all bicrossed products between the Taft algebra and the group Hopf algebra $K[D_{2n}]$, where $D_{2n}$ denotes the dihedral group.

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arXiv2018

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.

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arXiv2017

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.

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arXiv2016

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 Hölder'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.

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arXiv2016

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 $n-1$.

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arXiv2016

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 $π: E \to A$ a surjective linear map with $V = {\rm Ker} (π)$. All algebra structures on $E$ such that $π: E \to A$ 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.

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arXiv2015

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.

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arXiv2015

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.

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