The subgroup of generated using the adjoint action is called the inner automorphism group of . The group is denoted . These form a normal subgroup in the group of automorphisms, and the quotient is known as the outer automorphism group.[1]
Diagram automorphisms
It is known that the outer automorphism group for a simple Lie algebra is isomorphic to the group of diagram automorphisms for the corresponding Dynkin diagram in the classification of Lie algebras.[2] The only algebras with non-trivial outer automorphism group are therefore and .
Outer automorphism group
There are ways to concretely realize these automorphisms in the matrix representations of these groups. For , the automorphism can be realized as the negative transpose. For , the automorphism is obtained by conjugating by an orthogonal matrix in with determinant -1.
The set of derivations on a Lie algebra is denoted , and is a subalgebra of the endomorphisms on , that is . They inherit a Lie algebra structure from the Lie algebra structure on the endomorphism algebra, and closure of the bracket follows from the Leibniz rule.
For each in a Lie group , let denote the differential at the identity of the conjugation by . Then is an automorphism of , the adjoint action by .
Theorems
The Borel–Morozov theorem states that every solvable subalgebra of a complex semisimple Lie algebra can be mapped to a subalgebra of a Cartan subalgebra of by an inner automorphism of . In particular, it says that , where are root spaces, is a maximal solvable subalgebra (that is, a Borel subalgebra).[3]
References
^Humphreys 1972
^Humphreys 1972
^Serre 2000, Ch. VI, Theorem 5.
E. Cartan, Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sc. math. 49, 1925, pp. 361–374.
Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN978-3-540-67827-4.
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