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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.09824 |
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| _version_ | 1866910699270176768 |
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| author | Dokuchaev, Mikhailo Jerez, Emmanuel |
| author_facet | Dokuchaev, Mikhailo Jerez, Emmanuel |
| contents | 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 \( (Ω_σ, G, \hatθ) \), where \( Ω_σ\) is a compact, totally disconnected Hausdorff space, and \( σ\) acts as a twist for \( \hatθ \). We show that the twisted partial group algebra \( κ_{par}^σ G \) can be realized as a crossed product \( {\mathscr L}(Ω_σ) \rtimes_{(\hatθ, σ)} G \), with \( {\mathscr L}(Ω_σ) \) denoting the \( κ\)-algebra of locally constant functions \( Ω_σ\to κ\). The space \( Ω_σ\) corresponds to the spectrum of a unital commutative subalgebra in \( κ_{par}^σ G \), generated by idempotents. By describing \( Ω_σ\) as a subspace of the Bernoulli space \( 2^G \), we examine conditions under which the spectral partial action \( \hatθ \) is topologically free, impacting the ideal structure of \( κ_{par}^σ G \).
We further explore generating idempotent factor sets of \( G \) and present conditions on them to ensure the topological freeness of \( \hatθ \). Inspired by Exel's semigroup \( \mathcal{S}(G) \), which governs partial actions and representations of \( G \) and relates to \( κ_{par}G \), we characterize the twisted partial group algebra \( κ_{par}^σG \) as generated by a \( κ\)-cancellative inverse semigroup constructed from elements of \( Ω_σ\). When \( Ω_σ\) is discrete, we demonstrate that \( κ_{par}^σ G \) decomposes into a product of matrix algebras over twisted subgroup algebras, generalizing known results for finite \( G \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09824 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twisted partial group algebra and related topological partial dynamical system Dokuchaev, Mikhailo Jerez, Emmanuel Rings and Algebras Primary 16S35, Secondary 20C25 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 \( (Ω_σ, G, \hatθ) \), where \( Ω_σ\) is a compact, totally disconnected Hausdorff space, and \( σ\) acts as a twist for \( \hatθ \). We show that the twisted partial group algebra \( κ_{par}^σ G \) can be realized as a crossed product \( {\mathscr L}(Ω_σ) \rtimes_{(\hatθ, σ)} G \), with \( {\mathscr L}(Ω_σ) \) denoting the \( κ\)-algebra of locally constant functions \( Ω_σ\to κ\). The space \( Ω_σ\) corresponds to the spectrum of a unital commutative subalgebra in \( κ_{par}^σ G \), generated by idempotents. By describing \( Ω_σ\) as a subspace of the Bernoulli space \( 2^G \), we examine conditions under which the spectral partial action \( \hatθ \) is topologically free, impacting the ideal structure of \( κ_{par}^σ G \). We further explore generating idempotent factor sets of \( G \) and present conditions on them to ensure the topological freeness of \( \hatθ \). Inspired by Exel's semigroup \( \mathcal{S}(G) \), which governs partial actions and representations of \( G \) and relates to \( κ_{par}G \), we characterize the twisted partial group algebra \( κ_{par}^σG \) as generated by a \( κ\)-cancellative inverse semigroup constructed from elements of \( Ω_σ\). When \( Ω_σ\) is discrete, we demonstrate that \( κ_{par}^σ G \) decomposes into a product of matrix algebras over twisted subgroup algebras, generalizing known results for finite \( G \). |
| title | Twisted partial group algebra and related topological partial dynamical system |
| topic | Rings and Algebras Primary 16S35, Secondary 20C25 |
| url | https://arxiv.org/abs/2411.09824 |