Abstract fractional linear transformations
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911494822690816 |
|---|---|
| author | Handelman, David |
| author_facet | Handelman, David |
| contents | We begin with (densely-defined) fractional linear transformations (FLT) on (some) Banach algebras and their relatives. This leads to Wedderburn's continued fractions (recursively-defined noncommutative polynomials) for any ring. Along the way, we discover a one-parameter family of (noncommutative) polynomials \st if one of them is invertible, then read in the opposite order, the corresponding polynomial is also invertible (extending the well known $1+ab$ is invertible if $1+ ba$ is, and the not-so-well-known, $a + abc + c$ and $a + cba + c$).
This in turn leads to a definition of FLT for general rings $R$, which turns out to be PE$(2,R)$ (the projective elementary group). Using Wedderburn's polynomials, this permits us to define a length function on PE$(2,R)$, which suggests a stable range type condition (for $n =1$, it {\it is\/} stable range one, but higher values do not correspond.
Again using the length results, we prove the expected results for PE$(2,R)$: under very modest conditions on $R$, the commutator subgroup of PE$(2,R)$ is perfect and of index one or two.
Along the same lines, we also prove results on simplicity of the commutator subgroup: we require the usual generative properties on the simple ring $R$, as well either the very restrictive $1$ in the range, or a mild condition about invertibles, involving intersections of three translates of GL$(1,R)$. This last property is explored in the appendices, which give examples (and non-examples).
Numerous questions suggest themselves throughout. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17383 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Abstract fractional linear transformations Handelman, David Functional Analysis Rings and Algebras (primary) 16U99 (secondary) 11T30, 19B10, 20H25, 46B99 We begin with (densely-defined) fractional linear transformations (FLT) on (some) Banach algebras and their relatives. This leads to Wedderburn's continued fractions (recursively-defined noncommutative polynomials) for any ring. Along the way, we discover a one-parameter family of (noncommutative) polynomials \st if one of them is invertible, then read in the opposite order, the corresponding polynomial is also invertible (extending the well known $1+ab$ is invertible if $1+ ba$ is, and the not-so-well-known, $a + abc + c$ and $a + cba + c$). This in turn leads to a definition of FLT for general rings $R$, which turns out to be PE$(2,R)$ (the projective elementary group). Using Wedderburn's polynomials, this permits us to define a length function on PE$(2,R)$, which suggests a stable range type condition (for $n =1$, it {\it is\/} stable range one, but higher values do not correspond. Again using the length results, we prove the expected results for PE$(2,R)$: under very modest conditions on $R$, the commutator subgroup of PE$(2,R)$ is perfect and of index one or two. Along the same lines, we also prove results on simplicity of the commutator subgroup: we require the usual generative properties on the simple ring $R$, as well either the very restrictive $1$ in the range, or a mild condition about invertibles, involving intersections of three translates of GL$(1,R)$. This last property is explored in the appendices, which give examples (and non-examples). Numerous questions suggest themselves throughout. |
| title | Abstract fractional linear transformations |
| topic | Functional Analysis Rings and Algebras (primary) 16U99 (secondary) 11T30, 19B10, 20H25, 46B99 |
| url | https://arxiv.org/abs/2511.17383 |