Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 245Root Systems and Lie TheoryIrreducible root systems classify a number of related objects in Lie theory,notably:(1) Simple complex Lie algebras;(2) Simple complex Lie groups;(3) Simply connected complex Lie groups which are simple modulo centers;and(4) Simple compact Lie groups.In each case, the roots are non–zero weights of the adjoint representation.A root system can also be said to describe a plant’s root and associatedsystems.3.8.6.4 Simple and Semisimple Lie Groups and AlgebrasA simple Lie group is a Lie group which is also a simple group. Thesegroups, and groups closely related to them, include many of the so–calledclassical groups of geometry, which lie behind projective geometry and othergeometries derived from it by the Erlangen programme of Felix Klein. Theyalso include some exceptional groups, that were first discovered by thosepursuing the classification of simple Lie groups. The exceptional groupsaccount for many special examples and configurations in other branchesof mathematics. In particular the classification of finite simple groups dependedon a thorough prior knowledge of the ‘exceptional’ possibilities.The complete listing of the simple Lie groups is the basis for the theoryof the semisimple Lie groups and reductive groups, and their representationtheory. This has turned out not only to be a major extension of the theoryof compact Lie groups (and their representation theory), but to be of basicsignificance in mathematical physics.Such groups are classified using the prior classification of the complexsimple Lie algebras. It has been shown that a simple Lie group has a simpleLie algebra that will occur on the list given there, once it is complexified(that is, made into a complex vector space rather than a real one). Thisreduces the classification to two further matters.The groups SO(p, q, R) and SO(p + q, R), for example, give rise to differentreal Lie algebras, but having the same Dynkin diagram. In generalthere may be different real forms of the same complex Lie algebra.Secondly, the Lie algebra only determines uniquely the simply connected