Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 243is admissible.(2) An admissible diagram has no loops.(3) No node has more than three lines attached to it.(4) A sequence of nodes with only two single lines can be collapsed to givean admissible diagram.(5) The only connected diagram with a triple line has two nodes.A Coxeter–Dynkin diagram, also called a Coxeter graph, is the same asa Dynkin diagram, but without the arrows. The Coxeter diagram is sufficientto characterize the algebra, as can be seen by enumerating connecteddiagrams.The simplest way to recover a simple Lie algebra from its Dynkin diagramis to first reconstruct its Cartan matrix (A ij ). The ith node andjth node are connected by A ij A ji lines. Since A ij = 0 iff A ji = 0, andotherwise A ij ∈ {−3, −2, −1}, it is easy to find A ij and A ji , up to order,from their product. The arrow in the diagram indicates which is larger.For example, if node 1 and node 2 have two lines between them, from node1 to node 2, then A 12 = −1 and A 21 = −2.However, it is worth pointing out that each simple Lie algebra can beconstructed concretely. For instance, the infinite families A n , B n , C n , andD n correspond to the special linear Lie algebra gl(n+1, C), the odd orthogonalLie algebra so(2n + 1, C), the symplectic Lie algebra sp(2n, C), andthe even orthogonal Lie algebra so(2n, C). The other simple Lie algebrasare called exceptional Lie algebras, and have constructions related to theoctonions.To prove this classification Theorem, one uses the angles between pairsof roots to encode the root system in a much simpler combinatorial object,the Dynkin diagram. The Dynkin diagrams can then be classified accordingto the scheme given above.To every root system is associated a corresponding Dynkin diagram.Otherwise, the Dynkin diagram can be extracted from the root system bychoosing a base, that is a subset ∆ of Φ which is a basis of V with thespecial property that every vector in Φ when written in the basis ∆ haseither all coefficients ≥ 0 or else all ≤ 0.The vertices of the Dynkin diagram correspond to vectors in ∆. Anedge is drawn between each non–orthogonal pair of vectors; it is a doubleedge if they make an angle of 135 degrees, and a triple edge if they makean angle of 150 degrees. In addition, double and triple edges are markedwith an angle sign pointing toward the shorter vector.