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Mikhailo Dokuchaev

Publications and source records attributed to Mikhailo Dokuchaev.

At least 19 recordsLinked to original sources
arXiv2026

Globalization of Partial Group Actions on Not Necessarily Associative Algebras and Covariant Representations

We extend the concept of a partial group action to non-associative algebras in a variety \(\mathcal{V}(I)\), solve the globalization problem within \(\mathcal{V}(I)\) and examine its universal property. It is achieved using what we call the ``$\Lambda$-construction'', which we also apply to deal with covariant representations in the associative and Lie algebra settings, considering related categories and constructing an adjoint pair of functors between them. We also show that the $\Lambda$-construction behaves well with semidirect products of Lie algebras.

math.RA↗
arXiv2026

Graded Equivalence for Graded Idempotent Rings

In this paper, we extend the study of graded equivalences to the case of general idempotent graded rings. We prove that the existence of a graded equivalence between two categories of graded torsion-free unital modules may be characterized by the existence of a Morita context with surjective trace maps. As an application of our results we relate certain lattices of graded submodules and graded ideals of graded equivalent garded rings and give some properties invariant under graded equivalences.

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arXiv2025

Homology and cohomology of crossed products by inverse monoid actions and Steinberg algebras

Given a unital action $\theta $ of an inverse monoid $S$ on an algebra $A$ over a filed $K$ we produce (co)homology spectral sequences which converge to the Hochschild (co)homology of the crossed product $A\rtimes_\theta S$ with values in a bimodule over $A\rtimes_\theta S$. The spectral sequences involve a new kind of (co)homology of the inverse monoid $S,$ which is based on $KS$-modules. The spectral sequences take especially nice form, when $(A\rtimes_\theta S)^e $ is flat as a left (homology case) or right (cohomology case) $A^e$-module, involving also the Hochschild (co)homology of $A.$ Same nice spectral sequences are also obtained if $K$ is a commutative ring, over which $A$ is projective, and $S$ is $E$-unitary. We apply our results to the Steinberg algebra $A_K(\mathscr{G})$ over a field $K$ of an ample groupoid $\mathscr{G},$ whose unit space $\mathscr{G} ^{(0)}$ is compact. In the homology case our spectral sequence collapses on the $p$-axis, resulting in an isomorphism between the Hochschild homology of $A_K(\mathscr{G})$ with values in an $A_K(\mathscr{G})$-bimodule $M$ and the homology of the inverse semigroup of the compact open bisections of $\mathscr{G}$ with values in the invariant submodule of $M.$

math.RA↗
arXiv2025

Inverse semialgebras and partial actions of Lie algebras

We introduce the concept of a non-associative (i.e. non-necessarily associtive) inverse semialgebra over a field, the Lie version of which is inspired by the set of all partially defined derivations of a non-associative algebra, whereas the associative case is based on such examples as the set of all partially defined linear maps of a vector space, the set of all sections of the structural sheaf of a scheme, the set of all regular functions defined on open subsets of an algebraic variety and the set of all smooth real valued functions defined on open subsets of a smooth manifold. Given a Lie algebra $L$ we define the notion of a partial action of $L$ on a non-associative algebra $A$ as an appropriate premorphism and introduce a Lie inverse semialgebra $E(L),$ which is a Lie analogue of R. Exel's inverse semigroup $S(G)$ that governs the partial actions of a group $G.$ We discuss how $E(L)$ controls the premorphisms from $L$ to $A,$ obtaining results on its total control. We define the concept of an $F$-inverse Lie semialgebra and obtain Lie theoretic analogues of some classical results of the theory of inverse semigroups, namely, we show that the category of partial representations of $L$ in meet semilattices is equivalent to the category ${\mathcal F}$ of $F$-inverse Lie semialgebras with morphisms that preserve the greatest elements of $\sigma$-classes. In addition, we establish an adjunction between the category of Lie algebras and the category ${\mathcal F}.$

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arXiv2024

Twisted partial group algebra and related topological partial dynamical system

Given a group \( G \), a field \( \kappa \), and a factor set \( \sigma \) arising from a partial projective \( \kappa \)-representation of \( G \). This leads to the construction of a topological partial dynamical system \( (\Omega_\sigma, G, \hat{\theta}) \), where \( \Omega_\sigma \) is a compact, totally disconnected Hausdorff space, and \( \sigma \) acts as a twist for \( \hat{\theta} \). We show that the twisted partial group algebra \( \kappa_{par}^{\sigma} G \) can be realized as a crossed product \( {\mathscr L}(\Omega_\sigma) \rtimes_{(\hat{\theta}, \sigma)} G \), with \( {\mathscr L}(\Omega_\sigma) \) denoting the \( \kappa \)-algebra of locally constant functions \( \Omega_\sigma \to \kappa \). The space \( \Omega_\sigma \) corresponds to the spectrum of a unital commutative subalgebra in \( \kappa_{par}^{\sigma} G \), generated by idempotents. By describing \( \Omega_\sigma \) as a subspace of the Bernoulli space \( 2^G \), we examine conditions under which the spectral partial action \( \hat{\theta} \) is topologically free, impacting the ideal structure of \( \kappa_{par}^{\sigma} G \). We further explore generating idempotent factor sets of \( G \) and present conditions on them to ensure the topological freeness of \( \hat{\theta} \). Inspired by Exel's semigroup \( \mathcal{S}(G) \), which governs partial actions and representations of \( G \) and relates to \( \kappa_{par}G \), we characterize the twisted partial group algebra \( \kappa_{par}^{\sigma}G \) as generated by a \( \kappa \)-cancellative inverse semigroup constructed from elements of \( \Omega_\sigma \). When \( \Omega_\sigma \) is discrete, we demonstrate that \( \kappa_{par}^{\sigma} G \) decomposes into a product of matrix algebras over twisted subgroup algebras, generalizing known results for finite \( G \).

math.RA↗
arXiv2024

Partial generalized crossed products, Brauer groups and a comparison of seven-term exact sequences

Given a unital partial action $\alpha $ of a group $G$ on a commutative ring $R$ we denote by $ {\bf PicS} _{R^{\alpha}}(R) $ the Picard monoid of the isomorphism classes of partially invertible $R$-bimodules, which are central over the subring $R^{\alpha} \subseteq R$ of $\alpha$-invariant elements, and consider a specific unital partial representation $\Theta : G \to {\bf PicS} _{R^{\alpha}}(R), $ along with the abelian group $\mathcal {C}(\Theta/R)$ of the isomorphism classes of partial generalized crossed products related to $\Theta,$ which already showed their importance in obtaining a partial action analogue of the Chase-Harrison-Rosenberg seven-term exact sequence. We give a description of $\mathcal {C}(\Theta/R)$ in terms partial generalized products of the form $\mathcal D(f \Theta)$ where $f$ is partial $1$-cocycle of $G$ with values in a submonoid of $ {\bf PicS}_{R^{\alpha}}(R).$ Assuming that $G$ is finite and that $R^{\alpha} \subseteq R$ is a partial Galois extension, we prove that any Azumaya $R^\alpha$-algebra, containing $R$ as a maximal commutative subalgebra, is isomorphic to a partial generalized crossed product. Furthermore, we show that the relative Brauer group $\mathcal B(R/R^\alpha)$ can be seen as a quotient of $\mathcal {C}(\Theta/R)$ by a subgroup isomorphic to the Picard group of $R.$ Finally, we prove that the analogue of the Chase-Harrison-Rosenberg sequence, obtained earlier for partial Galois extensions of commutative rings, can be derived from a recent seven-term exact sequence established in a non-commutative setting.

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arXiv2023

(Co)Homology of Partial Smash Products

Given a cocommutative Hopf algebra $\mathcal{H}$ over a commutative ring $K$ and a symmetric partial action of $\mathcal{H}$ on a $K$-algebra $A,$ we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product $A \# \mathcal{H},$ involving the Hochschild homology of $A$ and the partial homology of $\mathcal{H}.$ An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of $\mathcal{H}$ under consideration is based on the category of the partial representations of $\mathcal{H}.$ A specific partial representation of $\mathcal{H}$ on a subalgebra $\mathcal{B}$ of the partial ``Hopf" algebra $\mathcal{H}_{par} $ is involved in the definition and we construct a projective resolution of $\mathcal{B}.$

math.RA↗
arXiv2023

The twisted partial group algebra and (co)homology of partial crossed products

Given a group $G$ and a partial factor set $\sigma $ of $G,$ we introduce the twisted partial group algebra $\kappa_{par}^{\sigma}G,$ which governs the partial projective $\sigma$-representations of $G$ into algebras over a filed $\kappa.$ Using the relation between partial projective representations and twisted partial actions we endow $\kappa_{par}^\sigma G$ with the structure of a crossed product by a twisted partial action of $G$ on a commutative subalgebra of $\kappa_{par}^{\sigma} G.$ Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product $A\ast_{\Theta} G,$ involving the Hochschild homology of $A$ and the partial homology of $G,$ where ${\Theta}$ is a unital twisted partial action of $G$ on a $\kappa$-algebra $A$ with a $\kappa $-based twist. An analogous third quadrant cohomological spectral sequence is also obtained.

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arXiv2021

Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups

We define and study the notion of a crossed module over an inverse semigroup and the corresponding $4$-term exact sequences, called crossed module extensions. For a crossed module $A$ over an $F$-inverse monoid $T$, we show that equivalence classes of admissible crossed module extensions of $A$ by $T$ are in a one-to-one correspondence with the elements of the cohomology group $H^3_\le(T^1,A^1)$.

math.GR↗
arXiv2021

The cone of quasi-semimetrics and exponent matrices of tiled orders

Finite quasi semimetrics on $n$ can be thought of as nonnegative valuations on the edges of a complete directed graph on $n$ vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were studied for small $n$ by Deza, Dutour and Panteleeva. We show that the symmetry and combinatorial symmetry groups are as they conjectured. Integral quasi semimetrics have apecial place in the theory of tiled orders, being known as exponent matrices, and can be viewed as monoids under componentwise maximum; we provide a novel derivation of the automorphism group of that monoid. Some of these results follow from more general consideration of polyhedral cones that are closed under componentwise maximum.

math.CO↗
arXiv2021

Partial generalized crossed products and a seven term exact sequence (expanded version)

Given a non-necessarily commutative unital ring $R$ and a unital partial representation $\Theta $ of a group $G$ into the Picard semigroup $\mathbf{PicS} (R)$ of the isomorphism classes of partially invertible $R$-bimodules, we construct an abelian group $\mathcal{C}(\Theta /R) $ formed by the isomorphism classes of partial generalized crossed products related to $\Theta $ and identify an appropriate second partial cohomology group of $G$ with a naturally defined subgroup $\mathcal{C}_0(\Theta /R) $ of $\mathcal{C}(\Theta /R).$ Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings $R\subseteq S$ with the same unity and a unital partial representation $ G \to \mathcal{S}_R(S)$ of an arbitrary group $G$ into the monoid $\mathcal{S}_R(S)$ of the $R$-subbimodules of $S.$ This generalizes the works by Kanzaki and Miyashita.

math.RA↗
arXiv2020

Globalization of partial cohomology of groups

We study the relations between partial and global group cohomology with values in a commutative unital ring $\mathcal{A}$. In particular, for a unital partial action of a group $G$ on $\mathcal{A}$, such that $\mathcal{A}$ is a direct product of commutative indecomposable rings, we show that any partial $n$-cocycle of $G$ with values in $\mathcal{A}$ is globalizable.

math.RA↗
arXiv2020

Homology and cohomology via the partial group algebra

We study partial homology and cohomology from ring theoretic point of view via the partial group algebra $\mathbb{K}_{par}G$. In particular, we link the partial homology and cohomology of a group $G$ with coefficients in an irreducible (resp. indecomposable) $\mathbb{K}_{par}G$-module with the ordinary homology and cohomology groups of $G$ with in general non-trivial coefficients. Furthermore, we compare the standard cohomological dimension $cd_{ \ \mathbb{K}}(G)$ (over a field $\mathbb{K}$) with the partial cohomological dimension $cd_{ \ \mathbb{K}}^{par}(G)$ (over $\mathbb{K}$) and show that $cd_{ \ \mathbb{K}}^{par}(G) \geq cd_{ \ \mathbb{K}}(G)$ and that there is equality for $G = \mathbb{Z}$.

math.GR↗
arXiv2019

Globalization of group cohomology in the sense of Alvares-Alves-Redondo

Recently E. R. Alvares, M. M. Alves and M. J. Redondo introduced a cohomology for a group $G$ with values in a module over the partial group algebra $K_{\mathrm{par}}(G)$, which is different from the partial group cohomology defined earlier by the first two named authors of the present paper. Given a unital partial action $α$ of $G$ on a (unital) algebra $\mathcal{A}$ we consider $\mathcal{A}$ as a $K_{\mathrm{par}}(G)$-module in a natural way and study the globalization problem for the cohomology in the sense of Alvares-Alves-Redondo with values in $\mathcal{A}$. The problem is reduced to an extendibility property of cocycles. Furthermore, assuming that $\mathcal{A}$ is a product of blocks, we prove that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous. As a consequence we obtain that the Alvares-Alves-Redondo cohomology group $H_{par}^n(G,\mathcal{A})$ is isomorphic to the usual cohomology group $H^n(G,\mathcal{M}(\mathcal{B}))$, where $\mathcal{M}(\mathcal{B})$ is the multiplier algebra of $\mathcal{B}$ and $\mathcal{B}$ is the algebra under the enveloping action of $α$.

math.RA↗
arXiv2019

Realizing corners of Leavitt path algebras as Steinberg algebras, with corresponding connections to graph $C^*$-algebras

We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra $L_K(E)$ of an arbitrary graph $E$ with coefficients in a field $K$ is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every $K$-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a $C^*$-algebra of a countable graph is isomorphic to the $C^*$-algebra of an ample groupoid.

math.RA↗
arXiv2019

Partial actions and proper extensions of two-sided restriction semigroups

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study several classes of partial actions of two-sided restriction semigroups that generalize partial actions of monoids and of inverse semigroups. We establish an adjunction between the category ${\mathcal{P}}(S)$ of proper extensions of a restriction semigroup (or, in particular, an inverse semigroup) $S$ and a category ${\mathcal{A}}(S)$ of partial actions of $S$ subject to certain conditions going back to the work of O'Carroll. In the category ${\mathcal{A}}(S)$, we specify two isomorphic subcategories, one being reflective and the other one coreflective, each of which is equivalent to the category ${\mathcal{P}}(S)$.

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