Adjoint representation of a Lie group

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or algebraic group

The linear representation of in the tangent space (or in the Lie algebra of ) mapping each to the differential of the inner automorphism . If is a linear group in a space , then

The kernel contains the centre of , and if is connected and if the ground field has characteristic zero, coincides with this centre. The differential of the adjoint representation of at coincides with the adjoint representation of .

The adjoint representation of a Lie algebra is the linear representation of the algebra into the module acting by the formula

where is the bracket operation in the algebra . The kernel is the centre of the Lie algebra . The operators are derivations of and are called inner derivations. The image is called the adjoint linear Lie algebra and is an ideal in the Lie algebra of all derivations of , moreover is the one-dimensional cohomology space of , defined by the adjoint representation. In particular, if is a semi-simple Lie algebra over a field of characteristic zero.


[1] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[4] J.E. Humphreys, "Linear algebraic groups" , Springer (1975)



[a1] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French)
How to Cite This Entry:
Adjoint representation of a Lie group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article