Ivancevic_Applied-Diff-Geom

242 Applied Differential Geometry: A Modern IntroductionDynkin DiagramsA Dynkin diagram is a graph with a few different kinds of possible edges(see Figure 3.10). The connected components of the graph correspond tothe irreducible subalgebras of g. So a simple Lie algebra’s Dynkin diagramhas only one component. The rules are restrictive. In fact, there are onlycertain possibilities for each component, corresponding to the classificationof semi–simple Lie algebras (see, e.g., [Conway et al. (1985)]).Fig. 3.10 The problem of classifying irreducible root systems reduces to the problem ofclassifying connected Dynkin diagrams.The roots of a complex Lie algebra form a lattice of rank k in a Cartansubalgebra h ⊂ g, where k is the Lie algebra rank of g. Hence, the rootlattice can be considered a lattice in R k . A vertex, or node, in the Dynkindiagram is drawn for each Lie algebra simple root, which corresponds toa generator of the root lattice. Between two nodes α and β, an edge isdrawn if the simple roots are not perpendicular. One line is drawn if theangle between them is 2π/3, two lines if the angle is 3π/4, and three linesare drawn if the angle is 5π/6. There are no other possible angles betweenLie algebra simple roots. Alternatively, the number of lines N between thesimple roots α and β is given byN = A αβ A βα =2 〈α, β〉|α| 2 2 〈β, α〉|β| 2 = 4 cos 2 θ,where A αβ = 2〈α,β〉|α|is an entry in the Cartan matrix (A 2αβ ) (for details onCartan matrix see, e.g., [Helgason (2001); Weisstein (2004)]). In a Dynkindiagram, an arrow is drawn from the longer root to the shorter root (whenthe angle is 3π/4 or 5π/6).Here are some properties of admissible Dynkin diagrams:(1) A diagram obtained by removing a node from an admissible diagram