Automorphism of a Lie algebra

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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

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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

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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.

Derivations

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A derivation on a Lie algebra is a linear map   satisfying the Leibniz rule   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.

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

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  • 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

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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

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  • 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.