The Carlitz Algebras
| dc.creator | Bavula, V. V. | |
| dc.date | 2005-05-18 | |
| dc.date.accessioned | 2026-07-07T05:20:02Z | |
| dc.date.available | 2026-07-07T05:20:02Z | |
| dc.description | The Carlitz $\mathbb{F}_q$-algebra $C=C_ν$, $ν\in \mathbb{N}$, is generated by an algebraically closed field $\CK $ (which contains a non-discrete locally compact field of positive characteristic $p>0$, i.e. $K\simeq \mathbb{F}_q[[ x,x^{-1}]]$, $q=p^ν$), by the (power of the) {\em Frobenius} map $X=X_ν:f\mapsto f^q$, and by the {\em Carlitz derivative} $Y=Y_ν$. It is proved that the Krull and global dimensions of $C$ are 2, a classification of simple $C$-modules and ideals are given, there are only {\em countably many} ideals, they commute $(IJ=JI)$, and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple $C$-module is a sum of eigenspaces of the element $YX$ (the set of eigenvalues for $YX$ is given explicitly for each simple $C$-module). This fact is crucial in finding the group $\Aut_{\Fq}(C)$ of $\Fq$-algebra automorphisms of $C$ and in proving that two distinct Carlitz rings are not isomorphic $(C_ν\not\simeq C_μ$ if $ν\neq μ$). The centre of $C$ is found explicitly, it is a UFD that contains {\em countably many} elements. | |
| dc.description | 16 pages | |
| dc.identifier | https://arxiv.org/abs/math/0505397 | |
| dc.identifier | http://arxiv.org/abs/math/0505397 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75241 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Representation Theory | |
| dc.subject | 16G99, 16D30, 16P40, 16U70 | |
| dc.title | The Carlitz Algebras | |
| dc.type | text |