Dynkin Diagram - Automorphisms

Automorphisms

In addition to isomorphism between different diagrams, some diagrams also have self-isomorphisms or "automorphisms". Diagram automorphisms correspond to outer automorphisms of the Lie algebra, meaning that the outer automorphism group Out = Aut/Inn equals the group of diagram automorphisms.

The diagrams that have non-trivial automorphisms are A_n, D_n, and E₆. In all these cases except for D₄, there is a single non-trivial automorphism (Out = C₂, the cyclic group of order 2), while for D₄, the automorphism group is the symmetric group on three letters (S₃, order 6) – this phenomenon is known as "triality". It happens that all these diagram automorphisms can be realized as Euclidean symmetries of how the diagrams are conventionally drawn in the plane, but this is just an artifact of how they are drawn, and not intrinsic structure.

For A_n, the diagram automorphism is reversing the diagram, which is a line. The nodes of the diagram index the fundamental weights, which (for A_n−1) are for, and the diagram automorphism corresponds to the duality Realized as the Lie algebra the outer automorphism can be expressed as negative transpose, which is how the dual representation acts.

For D_n, the diagram automorphism is switching the two nodes at the end of the Y, and corresponds to switching the two chiral spin representations. Realized as the Lie algebra the outer automorphism can be expressed as conjugation by a matrix in O(2n) with determinant −1. Note that so their automorphisms agree, while which is disconnected, and the automorphism corresponds to switching the two nodes.

For D₄, the fundamental representation is isomorphic to the two spin representations, and the resulting symmetric group on three letter (S₃, or alternatively the dihedral group of order 6, Dih₃) corresponds both to automorphisms of the Lie algebra and automorphisms of the diagram.

The automorphism group of E₆ corresponds to reversing the diagram, and can be expressed using Jordan algebras.

Disconnected diagrams, which correspond to semisimple Lie algebras, may have automorphisms from exchanging components of the diagram.

In positive characteristic there are additional diagram automorphisms – roughly speaking, in characteristic p one is allowed to ignore the arrow on bonds of multiplicity p in the Dynkin diagram when taking diagram automorphisms. Thus in characteristic 2 there is an order 2 automorphism of and of F₄, while in characteristic 3 there is an order 2 automorphism of G₂.