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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.16093 |
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| _version_ | 1866913244151545856 |
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| author | Michel, Manujith K. Srinivasan, Varadharaj R. |
| author_facet | Michel, Manujith K. Srinivasan, Varadharaj R. |
| contents | Let $F$ be a $δ-$field (differential field) of characteristic zero with an algebraically closed field of constants $F^δ$, $A$ be a $δ-F-$central simple algebra, $K$ be a Picard-Vessiot extension for the $δ-F-$module $A$ and $\mathscr G(K|F)$ be the $δ-$Galois group of $K$ over $F.$ We prove that a $δ-$field extension $L$ of $F,$ having $F^δ$ as its field of constants, splits the $δ-F-$central simple algebra $A$ if and only if the $δ-$field $K$ embeds in $L.$
We then extend the theory of $δ-F-$matrix algebras over a $δ-$field $F,$ put forward by Magid & Juan (2008), to arbitrary $δ-F-$central simple algebras. In particular, we establish a natural bijective correspondence between the isomorphism classes of $δ-F-$central simple algebras of dimension $n^2$ over $F$ that are split by the $δ-$field $K$ and the classes of inequivalent representations of the algebraic group $\mathscr G(K|F)$ in $\mathrm{PGL}_n(F^δ).$ We show that $\mathscr G(K|F)$ is a reductive or a solvable algebraic group if and only if $A$ has certain kinds of $δ-$right ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations Michel, Manujith K. Srinivasan, Varadharaj R. Rings and Algebras 12H05, 16KXX Let $F$ be a $δ-$field (differential field) of characteristic zero with an algebraically closed field of constants $F^δ$, $A$ be a $δ-F-$central simple algebra, $K$ be a Picard-Vessiot extension for the $δ-F-$module $A$ and $\mathscr G(K|F)$ be the $δ-$Galois group of $K$ over $F.$ We prove that a $δ-$field extension $L$ of $F,$ having $F^δ$ as its field of constants, splits the $δ-F-$central simple algebra $A$ if and only if the $δ-$field $K$ embeds in $L.$ We then extend the theory of $δ-F-$matrix algebras over a $δ-$field $F,$ put forward by Magid & Juan (2008), to arbitrary $δ-F-$central simple algebras. In particular, we establish a natural bijective correspondence between the isomorphism classes of $δ-F-$central simple algebras of dimension $n^2$ over $F$ that are split by the $δ-$field $K$ and the classes of inequivalent representations of the algebraic group $\mathscr G(K|F)$ in $\mathrm{PGL}_n(F^δ).$ We show that $\mathscr G(K|F)$ is a reductive or a solvable algebraic group if and only if $A$ has certain kinds of $δ-$right ideals. |
| title | Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations |
| topic | Rings and Algebras 12H05, 16KXX |
| url | https://arxiv.org/abs/2402.16093 |