nLab
Lorentz group

Contents

Definition

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

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 groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,

Revised on September 3, 2014 12:11:37 by Urs Schreiber (141.0.9.77)