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M. Dokuchaev

Publications and source records attributed to M. Dokuchaev.

13 recordsLinked to original sources
arXiv2025

Twisted Steinberg algebras, regular inclusions and induction

Given a field $K$ and an ample (not necessarily Hausdorff) groupoid $G$, we define the concept of a line bundle over $G$ inspired by the well known concept from the theory of C*-algebras. If $E$ is such a line bundle, we construct the associated twisted Steinberg algebra in terms of sections of $E$, extending the original construction introduced independently by Steinberg in 2010, and by Clark, Farthing, Sims and Tomforde in a 2014 paper (originally announced in 2011). We also generalize (strictly, in the non-Hausdorff case) the 2023 construction of (cocycle) twisted Steinberg algebras of Armstrong, Clark, Courtney, Lin, Mccormick and Ramagge. We then extend Steinberg's theory of induction of modules, not only to the twisted case, but to the much more general case of regular inclusions of algebras. Among our main results, we show that, under appropriate conditions, every irreducible module is induced by an irreducible module over a certain abstractly defined isotropy algebra. We also describe a process of disintegration of modules and use it to prove a version of the Effros-Hahn conjecture, showing that every primitive ideal coincides with the annihilator of a module induced from isotropy.

math.OA↗
arXiv2022

Strong equivalence of graded algebras

We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group $G$ is strongly-graded-equivalent to the skew group algebra by a product partial action of $G$. As to a more general idempotent graded algebra $B$, we point out that the Cohen-Montgomery duality holds for $B$, and $B$ is graded-equivalent to a global skew group algebra. We show that strongly-graded-equivalence preserves strong gradings and is nicely related to Morita equivalence of product partial actions. Furthermore, we prove that any product partial group action $\alpha $ is globalizable up to Morita equivalence; if such a globalization $\beta $ is minimal, then the skew group algebras by $\alpha $ and $\beta $ are graded-equivalent; moreover, $\beta $ is unique up to Morita equivalence. Finally, we show that strongly-graded-equivalent partially-strongly-graded algebras are stably isomorphic as graded algebras.

math.RA↗
arXiv2018

Partial cohomology of groups

We develop a cohomology theory of groups based on partial actions and explore its relation with the partial Schur multiplier as well as with cohomology of inverse semigroups.

math.GR↗
arXiv2016

The ideal structure of algebraic partial crossed products

Given a partial action of a discrete group $G$ on a Hausdorff, locally compact, totally disconnected topological space $X$, we consider the correponding partial action of $G$ on the algebra $L_c(X)$ consisting of all locally constant, compactly supported functions on $X$, taking values in a given field $K$. We then study the ideal structure of the algebraic partial crossed product $L_c(X)\rtimes G$. After developping a theory of induced ideals, we show that every ideal in $L_c(X)\rtimes G$ may be obtained as the intersection of ideals induced from isotropy groups, thus proving an algebraic version of the Effros-Hahn conjecture.

math.OA↗
arXiv2015

Partial actions and subshifts

Given a finite alphabet $Λ$, and a not necessarily finite type subshift $X\subseteq Λ^\infty$, we introduce a partial action of the free group $F(Λ)$ on a certain compactification $Ω_X$ of $X$, which we call the spectral partial action. The space $Ω_X$ has already appeared in many papers in the subject, arising as the spectrum of a commutative C*-algebra usually denoted by ${\cal D}_X$. Since the descriptions given of $Ω_X$ in the literature are often somewhat terse and obscure, one of our main goals is to present a sensible model for it which allows for a detailed study of its structure, as well as of the spectral partial action, from various points of view, including topological freeness and minimality. We then apply our results to study certain C*-algebras associated to $X$, introduced by Matsumoto and Carlsen. Most of the results we prove are already well known, but our proofs are hoped to be more natural and more in line with mainstream techniques used to treat similar C*-algebras. The clearer understanding of $Ω_X$ provided by our model in turn allows for a fine tuning of some of these results, including a necessary and sufficient condition for the minimality of the Carlsen-Matsumoto C*-algebra ${\cal O}_X$, generalizing a similar result of Thomsen.

math.OA↗
arXiv2013

Partial actions and automata

We use the notion of a partial action of a monoid to introduce a generalization of automata, which we call "a preautomaton". We study properties of preautomata and of languages recognized by preautomata.

cs.FL↗
arXiv2010

Globalization of twisted partial actions

Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.

math.RA↗
arXiv2008

Crossed products by twisted partial actions and graded algebras

For a twisted partial action Θof a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_ΘG is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_ΘG for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.

math.RA↗
arXiv2006

Associativity of crossed products by partial actions, enveloping actions and partial representations

Given a partial action αof a group G on an associative algebra A we consider the crossed product A x_αG. Using the algebras of multipliers of ideals of A we prove that A x_αG is associative, provided that all ideals of A are idempotent. This generalizes a previous result on the associativity of A x_αG in the context of C*-algebras. We also give a criteria for the existence of a global extension of a given partial action on an algebra and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with crossed product structure.

math.RA↗
arXiv2001

Imaginary Verma modules for the extended Affine Lie algebra $sl_2(C_q)$

We consider one of the most natural extended affine Lie lagebras, the algebra $sl_2({\mathbb C}_q)$ and begin a theory of its representations. In particular, we study a class of imaginary Verma modules, obtain a criterion of irreducibility and describe their submodule structure in "general position".

math.RT↗
arXiv1999

Partial Representations and Partial Group Algebras

The partial group algebra of a group G over a field K, denoted by K_{par}(G), is the algebra whose representations correspond to the partial representations of G over K-vector spaces. In this paper we study the structure of the partial group algebra K_{par}(G), where G is a finite group. In particular, given two finite abelian groups G_1 and G_2, we prove that if the characteristic of K is zero, then K_{par}(G_1) is isomorphic to K_{par}(G_2) if and only if G_1 is isomorphic to G_2.

math.GR↗