Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1906.05430

Sectional algebras of semigroupoid bundles

Luiz Gustavo Cordeiro
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^*$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions - via the construction of a sectional algebra - are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; Semidirect products of bundles correspond to "naïve" crossed products of algebras; Skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever $θ$ is a $\land$-preaction of a discrete inverse semigroupoid $S$ on an ample (possibly non-Hausdorff) groupoid $\mathcal{G}$, the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of $\mathcal{G}$ by $S$. This is a far-reaching generalization of analogous results which had been proven in particular cases.

Explore related subjects

math.RAmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Luiz Gustavo Cordeiro. 2019-06-13. Sectional algebras of semigroupoid bundles. https://arxiv.org/abs/1906.05430

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Graded identities for matrix algebras of order two over a finite field

Let $G$ be an arbitrary group and let $\mathbb{F}$ be a finite field. In this paper, we determine bases for the $T_G$-ideals of graded polynomial identities of the algebra $M_2(\mathbb{F})$ for all possible $G$-gradings. The bases obtained consist of finitely many non-trivial graded identities, and are finite whenever $G$ is finite.

math.RA↗
arXiv2026

Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras

Let $f:\N^{k}\to\N^{\ell}$ be a surjective monoid homomorphism and let $Γ$ be a row-finite $\ell$-graph with no sources and finitely many vertices. We give an explicit graded isomorphism from the Kumjian--Pask algebra of the pullback $f^{*}Γ$ onto the tensor product of $\KP_{\K}(Γ)$ and the group algebra of the kernel of the group completion of $f$. When $Γ$ is strongly aperiodic, but need not be cofinal, every maximal tail $T$ and every maximal ideal $\mathfrak m$ of the kernel group algebra determine an explicit primitive ideal and primitive quotient. If, in addition, $\K$ is uncountable and algebraically closed, these ideals exhaust the primitive spectrum. We prove that the resulting parametrisation is a homeomorphism for the product of the maximal-tail and Zariski topologies. Each primitive ideal is realised as the annihilator of a simple module induced from the isotropy of a path which is cofinal in $T$, and the character fibres are algebraic tori. Two examples exhibit, respectively, a single character fibre and the non-Hausdorff gluing of two such fibres.

math.RA↗
arXiv2026

A classification of group gradings on incidence algebras over commutative rings

Let $R$ be a commutative ring with 1, $P$ a locally finite partially ordered set, and $G$ a group. We derive necessary and sufficient conditions for an $R$-algebra isomorphism between the incidence algebra $I(P,R)$ and the group algebra $RG$. Then, for an indecomposable ring $R$, a finite poset $P$ and an arbitrary group $G$, we classify the $G$-gradings of $I(P,R)$ up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of $G$, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.

math.RA↗