Automorphism of a Lie algebra explained

In abstract algebra, an automorphism of a Lie algebra

ak g is an isomorphism from ak g to itself, that is, a bijective linear map preserving the Lie bracket. The set of automorphisms of ak{g} are denoted Aut ( ak{g}) , the automorphism group of ak{g} .

Inner and outer automorphisms

The subgroup of

\operatorname{Aut}( ak g) generated using the adjoint action e ^{\operatorname{ad} (x)}, x \in ak g is called the inner automorphism group of ak g . The group is denoted \operatorname{Aut} ^{0(ak{g})} . These form a normal subgroup in the group of automorphisms, and the quotient \operatorname{Aut}( ak{g})/\operatorname{Aut} ^{0(ak{g})} is known as the outer automorphism group.

Diagram automorphisms

It is known that the outer automorphism group for a simple Lie algebra

ak{g} is isomorphic to the group of diagram automorphisms for the corresponding Dynkin diagram in the classification of Lie algebras. The only algebras with non-trivial outer automorphism group are therefore A _{n} (n \geq 2 ), D _{n} and E _{6} .

ak g	Outer automorphism group
A _{n,} n \geq 2	Z _{2}
D _{n,} n ≠ 4	Z _{2}
D _{4}	S _{3}
E _{6}	Z _{2}

There are ways to concretely realize these automorphisms in the matrix representations of these groups. For

A _{n} = ak{sl}(n+1, C ) , the automorphism can be realized as the negative transpose. For D _{n} = ak{so}(2n) , the automorphism is obtained by conjugating by an orthogonal matrix in O( 2 n) with determinant -1.

Derivations

A derivation on a Lie algebra is a linear map$$\backslash delta:\; \backslash mathfrak\; \backslash rightarrow\; \backslash mathfrak$$satisfying the Leibniz rule$$\backslash delta[X,Y]\; =\; [\backslash delta\; X,\; Y]\; +\; [X,\; \backslash delta\; Y].$$The set of derivations on a Lie algebra

ak{g} is denoted \operatorname{der}( ak{g}) , and is a subalgebra of the endomorphisms on ak{g} , that is \operatorname{der}( ak{g}) < \operatorname{End}( ak{g}) . 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.\operatorname{ad}: ak{g} → \operatorname{End}( ak{g}) lies in \operatorname{der}( ak{g}) .

Through the Lie group-Lie algebra correspondence, the Lie group of automorphisms

\operatorname{Aut}( ak{g}) corresponds to the Lie algebra of derivations \operatorname{der}( ak{g}) .

For

ak{g} finite, all derivations are inner.

Examples

• For each

g in a Lie group G , let \operatorname{Ad} _{g} denote the differential at the identity of the conjugation by g . Then \operatorname{Ad} _{g} is an automorphism of ak{g} = \operatorname{Lie}(G) , the adjoint action by g .

See also: Cartan involution.

Theorems

The Borel–Morozov theorem states that every solvable subalgebra of a complex semisimple Lie algebra

ak g can be mapped to a subalgebra of a Cartan subalgebra ak h of ak g by an inner automorphism of ak g . In particular, it says that ak h ⊕ o plus _{\alpha} ak{g} _{\alpha} = : ak{h} ⊕ ak{g} ^{+} , where ak{g} _{\alpha} are root spaces, is a maximal solvable subalgebra (that is, a Borel subalgebra).

References

• E. Cartan, Le principe de dualité et la théorie des groupes simples et semi-simples. Bull. Sc. math. 49, 1925, pp. 361–374.
• Book: Humphreys, James . Introduction to Lie algebras and Representation Theory . James E. Humphreys . 1972 . Springer . 0387900535 . registration .
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