In abstract algebra, an automorphism of a Lie algebra is an isomorphism from to itself, that is, a bijective linear map preserving the Lie bracket. The set of automorphisms of are denoted , the automorphism group of .
Inner and outer automorphisms edit
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 edit
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 | |
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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.
Derivations edit
A derivation on a Lie algebra is a linear map
Due to the Jacobi identity, it can be shown that the image of the adjoint representation lies in .
Through the Lie group-Lie algebra correspondence, the Lie group of automorphisms corresponds to the Lie algebra of derivations .
For finite, all derivations are inner.
Examples edit
- 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 edit
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 edit
- ^ 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.
- Humphreys, James (1972). Introduction to Lie algebras and Representation Theory. Springer. ISBN 0387900535.
- Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4.