| Title:
|
Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops (English) |
| Author:
|
Lawson, Jimmie |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
319-331 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball $B$ in $n$-dimensional euclidean space $\mathbb R^n$. One notable achievement is a compact, convenient formula for the Möbius loop operation $a\ast b=(a+b)(1-ab)^{-1}$, where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w+z)(1+\overline wz)^{-1}$ for the Möbius loop multiplication in the unit complex disk). (English) |
| Keyword:
|
Bruck loop |
| Keyword:
|
Clifford algebra |
| Keyword:
|
gyrogroup |
| Keyword:
|
Möbius transformations |
| Keyword:
|
Vahlen matrices |
| Keyword:
|
involutive group |
| MSC:
|
15A66 |
| MSC:
|
20N05 |
| MSC:
|
51B10 |
| idZBL:
|
Zbl 1211.20064 |
| idMR:
|
MR2682484 |
| . |
| Date available:
|
2010-05-21T12:50:34Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140110 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
