Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops

Lawson, Jimmie. "Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 319-331. <http://eudml.org/doc/37763>.

@article{Lawson2010,
abstract = {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 $(w+z)(1+\overline\{w\}z)^\{-1\}$ for the Möbius loop multiplication in the unit complex disk).},
author = {Lawson, Jimmie},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Bruck loop; Clifford algebra; gyrogroup; Möbius transformations; Vahlen matrices; involutive group; Bruck loops; Möbius transformations; Clifford algebras; gyrogroups; Vahlen matrices; involutive groups},
language = {eng},
number = {2},
pages = {319-331},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops},
url = {http://eudml.org/doc/37763},
volume = {51},
year = {2010},
}

TY - JOUR
AU - Lawson, Jimmie
TI - Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2010
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 51
IS - 2
SP - 319
EP - 331
AB - 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 $(w+z)(1+\overline{w}z)^{-1}$ for the Möbius loop multiplication in the unit complex disk).
LA - eng
KW - Bruck loop; Clifford algebra; gyrogroup; Möbius transformations; Vahlen matrices; involutive group; Bruck loops; Möbius transformations; Clifford algebras; gyrogroups; Vahlen matrices; involutive groups
UR - http://eudml.org/doc/37763
ER -