Adjoint representation of a Lie group
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
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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
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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 0022.17104 |
[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 |
Adjoint representation of a Lie group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_representation_of_a_Lie_group&oldid=21787