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Adjoint representation of a Lie group

From Encyclopedia of Mathematics
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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.

References

[1] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0882.01025 Zbl 0534.22001 Zbl 0079.03903 Zbl 0058.26003 Zbl 0022.17104 Zbl 0016.20305 Zbl 0016.20304 Zbl 0015.24901
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[4] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039


Comments

References

[a1] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) pp. Chapt. 2; 3 (Translated from French) MR0682756 Zbl 0319.17002
How to Cite This Entry:
Adjoint representation of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_representation_of_a_Lie_group&oldid=21787
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article