The Carlitz algebras. (English) Zbl 1140.16009
Let \(K\) be a non-discrete locally compact field of positive characteristic \(p>0\). Then \(K\) has the form: \(K\) is a Laurent power series field in one variable \(x\) with coefficients in a finite field with \(q\) elements. Let \(\overline K\) be the algebraic closure of \(K\). Denote by \(\mathcal F\) the set of maps \(K\to\overline K\). The Carlitz algebra \(C\) is the subalgebra of all endomorphisms of \(\mathcal F\) over \(\mathbb{F}_q\) generated by the operators \(X(f)=f^q\) and by the operator \(Y=\root q\of\Delta\), where \(\Delta u(t)=u(xt)-xu(t)\), \(u(t)\in\mathcal F\). It is shown that \(C\) has global dimension 2. There is given a classification of simple \(C\)-modules and of ideals in \(C\). The group of automorphisms of \(C\) consists of automorphisms
\[
X\mapsto\alpha X,\quad Y\mapsto\gamma\root p\of{\alpha^{-1}}Y,\quad\mathcal K\ni k\mapsto \gamma k+\beta\in\mathcal K,
\]
where \(\alpha\in\mathcal K^*\), \(\gamma\in\mathbb{F}_q^*\), \(\delta\in\mathbb{F}_q\). These results are based on the observation that Carlitz algebras belong to the class of generalized Weyl algebras introduced by the author. Thus it is possible to apply the author’s previous results on the structure of these algebras and modules over them.
Reviewer: Vyacheslav A. Artamonov (Moskva)
MSC:
16S32 | Rings of differential operators (associative algebraic aspects) |
16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |
Keywords:
simple modules; Carlitz algebras; generalized Weyl algebras; groups of automorphisms; isomorphism problem; Krull dimension; global dimension; field extensionsReferences:
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