Editing Orthogonal group (section)

== Lie algebra ==
{{anchor|orthogonal Lie algebra}}{{anchor|special orthogonal Lie algebra}}
The [[Lie algebra]] corresponding to Lie groups {{math|O(''n'', ''F'' )}} and {{math|SO(''n'', ''F'' )}} consists of the [[skew-symmetric matrix|skew-symmetric]] {{math|''n'' × ''n''}} matrices, with the Lie bracket {{math|[ , ]}} given by the [[commutator]]. One Lie algebra corresponds to both groups. It is often denoted by <math>\mathfrak{o}(n, F)</math> or <math>\mathfrak{so}(n, F)</math>, and called the '''orthogonal Lie algebra''' or '''special orthogonal Lie algebra'''. Over real numbers, these Lie algebras for different {{math|''n''}} are the [[compact real form]]s of two of the four families of [[semisimple Lie algebra]]s: in odd dimension {{math|B<sub>''k''</sub>}}, where {{math|1=''n'' = 2''k'' + 1}}, while in even dimension {{math|D<sub>''r''</sub>}}, where {{math|1=''n'' = 2''r''}}.

Since the group {{math|SO(''n'')}} is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to ''ordinary'' representations of the orthogonal groups, and representations corresponding to ''projective'' representations of the orthogonal groups. (The projective representations of {{math|SO(''n'')}} are just linear representations of the universal cover, the [[spin group]] Spin(''n'').) The latter are the so-called [[spin representation]], which are important in physics.

More generally, given a vector space {{math|''V''}} (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form <math>\langle u,v \rangle</math>, the special orthogonal Lie algebra consists of tracefree endomorphisms <math>\varphi</math> which are skew-symmetric for this form (<math>\langle\varphi A, B\rangle = -\langle A, \varphi B\rangle</math>). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the [[bivector]]s of the [[exterior algebra]], the [[antisymmetric tensor]]s of  <math>\wedge^2 V</math>. The correspondence is given by:
: <math>v\wedge w \mapsto \langle v,\cdot\rangle w - \langle w,\cdot\rangle v</math>

This description applies equally for the indefinite special orthogonal Lie algebras <math>\mathfrak{so}(p, q)</math> for symmetric bilinear forms with signature {{math|(''p'', ''q'')}}.

Over real numbers, this characterization is used in interpreting the [[curl (mathematics)|curl]] of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.