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Dynkin Diagram


DynkinDiagrams

Every semisimple Lie algebra g is classified by its Dynkin diagram. A Dynkin diagram is a graph with a few different kinds of possible edges. The connected components of the graph correspond to the irreducible subalgebras of g. So a simple Lie algebra's Dynkin diagram has only one component. The rules are restrictive. In fact, there are only certain possibilities for each component, corresponding to the classification of semi-simple Lie algebras.

The roots of a complex Lie algebra form a lattice of rank k in a Cartan subalgebra h subset g, where k is the Lie algebra rank of g. Hence, the root lattice can be considered a lattice in R^k. A vertex, or node, in the Dynkin diagram is drawn for each Lie algebra simple root, which corresponds to a generator of the root lattice. Between two nodes alpha and beta, an edge is drawn if the simple roots are not perpendicular. One line is drawn if the angle between them is 2pi/3, two lines if the angle is 3pi/4, and three lines are drawn if the angle is 5pi/6. There are no other possible angles between Lie algebra simple roots. Alternatively, the number of lines N between the simple roots alpha and beta is given by

 N=A_(alphabeta)A_(betaalpha)=(2<alpha,beta>)/(|alpha|^2)(2<beta,alpha>)/(|beta|^2)=4cos^2theta,

where A_(alphabeta) is an entry in the Cartan matrix. In a Dynkin diagram, an arrow is drawn from the longer root to the shorter root (when the angle is 3pi/4 or 5pi/6).

DynkinDiagramG2Lattice

The picture above shows the two simple roots for G_2, at an angle of 5pi/6, in the root lattice. Therefore, the Dynkin diagram for G_2 has two nodes, with three lines between them.

Here are some properties of admissible Dynkin diagrams.

1. A diagram obtained by removing a node from an admissible diagram is 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 give an 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 as a Dynkin diagram, without the arrows, although sometimes these are also called Dynkin diagrams. The Coxeter diagram is sufficient to characterize the algebra, as can be seen by enumerating connected diagrams.

The simplest way to recover a simple Lie algebra from its Dynkin diagram is to first reconstruct its Cartan matrix (A_(ij)). The ith node and jth node are connected by A_(ij)A_(ji) lines. Since A_(ij)=0 iff A_(ji)=0, and otherwise A_(ij) in {-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 node 1 to node 2, then A_(12)=-1 and A_(21)=-2.

However, it is worth pointing out that each simple Lie algebra can be constructed concretely. For instance, the infinite families A_n, B_n, C_n, and D_n correspond to sl_(n+1)C the special linear Lie algebra, so_(2n+1)C the odd orthogonal Lie algebra, sp_(2n)C the symplectic Lie algebra, and so_(2n)C the even orthogonal Lie Algebra. The other simple Lie algebras are called exceptional Lie algebras, and have constructions related to the octonions.


See also

Cartan Matrix, Coxeter-Dynkin Diagram, Killing Form, Lie Algebra, Lie Group, Lie Algebra Root, Root Lattice, Simple Lie Algebra, Weyl Group

This entry contributed by Todd Rowland

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References

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, 1985.

Referenced on Wolfram|Alpha

Dynkin Diagram

Cite this as:

Rowland, Todd. "Dynkin Diagram." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DynkinDiagram.html

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