Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 239Basic complex Lie groups and their Lie algebras: 6LiegroupDescription Remarks Liealgb.C n group operationis additionAbelian, simplyconnected, notcompactAbelian, not simplyconnected,not compactsimply connected,not compact,for n = 1:isomorphic to C ×simple, semisimple,simply connected,for n ≥ 2:not compactnot connected,for n ≥ 2:not compactC × nonzero complexnumbers withmultiplicationGL(n, C) general lineargroup: invertiblen−by-n complexmatricesSL(n, C) special lineargroup: complexmatrices with determinant1O(n, C) orthogonalgroup: complexorthogonal matricessl(n, C) square matriceswith trace 0,with Lie bracketthe commutatorso(n, C) skew–symmetric squarecomplex matrices,with Liebracket the commutatorso(n, C) skew–symmetric squarecomplex matrices,with Liebracket the commutatorSO(n, C) special orthogonalgroup:complex orthogonalmatrices withdeterminant 1for n ≥ 2: notcompact,not simply connected,for n = 3and n ≥ 5: simpleand semisimpleC nCDescriptionthe Lie bracket iszerothe Lie bracket iszeroM(n, C) n−by-n matrices,with Lie bracketthe commutatordim/Cn1n 2n 2 −1n(n−1)/2n(n−1)/23.8.6.2 Representations of Lie groupsThe idea of a representation of a Lie group plays an important role in thestudy of continuous symmetry (see, e.g., [Helgason (2001)]). A great dealis known about such representations, a basic tool in their study being theuse of the corresponding ’infinitesimal’ representations of Lie algebras.Formally, a representation of a Lie group G on a vector space V (overa field K) is a group homomorphism G → Aut(V ) from G to the automorphismgroup of V . If a basis for the vector space V is chosen, therepresentation can be expressed as a homomorphism into GL(n, K). Thisis known as a matrix representation.6 The dimensions given are dimensions over C. Note that every complex Liegroup/algebra can also be viewed as a real Lie group/algebra of twice the dimension.