group theory

# Contents

## Definition

The Lorentz group is the orthogonal group for an invariant bilinear form of signature $\left(-+++\cdots \right)$, $O\left(n,1\right)$.

In physics the theory of special relativity the Lorentz group acts canonically as the group of linear isometries of Minkowski spacetime preserving a chosen basepoint. This is called the action by Lorentz transformations.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal group$\mathrm{O}\left(n\right)$Pin group$\mathrm{Pin}\left(n\right)$Tring group$\mathrm{Tring}\left(n\right)$
special orthogonal group$\mathrm{SO}\left(n\right)$Spin group$\mathrm{Spin}\left(n\right)$String group$\mathrm{String}\left(n\right)$
Lorentz group$\mathrm{O}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
anti de Sitter group$\mathrm{O}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\mathrm{Spin}\left(n,2\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
Narain group$O\left(n,n\right)$
Poincaré group$\mathrm{ISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$
super Poincaré group$\mathrm{sISO}\left(n,1\right)$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$$\phantom{\rule{thinmathspace}{0ex}}$

Revised on January 10, 2013 14:03:44 by Urs Schreiber (89.204.153.52)