arXiv subjects

Manujith K. Michel

Publications and source records attributed to Manujith K. Michel.

3 recordsLinked to original sources

arXiv2026

Iterative Derivations on Central Simple Algebras

We prove that an iterative derivation $δ_F$ on a field $F$ can be extended to an iterative derivation $δ_A$ on a central simple $F-$algebra $A$ if the characteristic of $F$ does not divide the exponent of $A$ in the Brauer group of $F.$ For a central simple $F-$algebra with an iterative derivation, we show the existence of a unique (up to isomorphism) Picard-Vessiot splitting field and from the nature its Galois group, we also describe the structure of the central simple algebra in terms of its $δ_A-$right ideals.

math.RA↗

arXiv2025

Extension of derivations to forms

The problem of extending derivations of a field $F$ to an $F-$algebra $B$ is widely studied in commutative algebra and non-commutative ring theory. For example, every derivation of $F$ extends to $B$ if $B$ is a separable algebraic extension or a central simple algebra over $F.$ We unify and generalize these results by showing that a derivation $d$ of $F$ with the field of constants $C$ extends to a finite dimensional algebra $B$ if $B$ is a form of some $C-$algebra having a smooth automorphism scheme $\rm G$. Furthermore, we show that the set of derivations of $B$ that extend the derivation $d$ of $F$ is in bijection with the set of derivations $δ$ such that $(Y,δ)$ is a differential $\rm G_F-$torsor where $Y$ is the $\rm G_F-$torsor corresponding to $B$.

math.RA↗

arXiv2024

Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations

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.

math.RA↗