Automorphism of a Lie algebra
Appearance
In abstract algebra, an automorphism of a Lie algebra is an isomorphism between and itself; i.e., a linear automorphism that preserves the bracket. The totality of them forms the automorphism group of . The subgroup of generated by matrix exponents is called the inner automorphism group of .
Examples
- 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 (i.e., a Borel subalgebra).[1]
References
- ^ 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, ISBN 978-3-540-67827-4.