2024/11/14 by Mikhailo Dokuchaev, Dokuchaev, Mikhailo, Emmanuel Jerez +1
Computer Science · Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 16S35 #Rings and Algebras (math.RA) #Secondary 20C25 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2411.09824
openalex publication_date 2024/11/14 · openalex created_date 2024/11/21 · openalex updated_date 2026/07/28
Given a group \( G \), a field \( κ\), and a factor set \( σ\) arising from a partial projective \( κ\)-representation of \( G \). This leads to the construction of a topological partial dynamical system \( (Ωσ, G, θ) \), where \( Ωσ\) is a compact, totally disconnected Hausdorff space, and \( σ\) acts as a twist for \( θ \). We show that the twisted partial group algebra \( κparσ G \) can be realized as a crossed product \( \mathscr L(Ωσ) \rtimes(θ, σ) G \), with \( \mathscr L(Ωσ) \) denoting the \( κ\)-algebra of locally constant functions \( Ωσ→ κ\). The space \( Ωσ\) corresponds to the spectrum of a unital commutative subalgebra in \( κparσ G \), generated by idempotents. By describing \( Ωσ\) as a subspace of the Bernoulli space \( 2G \), we examine conditions under which the spectral partial action \( θ \) is topologically free, impacting the ideal structure of \( κparσ G \). We further explore generating idempotent factor sets of \( G \) and present conditions on them to ensure the topological freeness of \( θ \). Inspired by Exel's semigroup \( S(G) \), which governs partial actions and representations of \( G \) and relates to \( κparG \), we characterize the twisted partial group algebra \( κparσG \) as generated by a \( κ\)-cancellative inverse semigroup constructed from elements of \( Ωσ\). When \( Ωσ\) is discrete, we demonstrate that \( κparσ G \) decomposes into a product of matrix algebras over twisted subgroup algebras, generalizing known results for finite \( G \).