- Adjoint representation
In
mathematics , the adjoint representation (or adjoint action) of aLie group "G" is the natural representation of "G" on its ownLie algebra . This representation is the linearized version of the action of "G" on itself by conjugation.Formal definition
Let "G" be a
Lie group and let be itsLie algebra (which we identify with "T""e""G", thetangent space to theidentity element in "G"). Define a
by the equation for all "g" in "G", where is theautomorphism group of "G" and theautomorphism is defined by:for all "h" in "G". It follows that the derivative of Ψ"g" at the identity is an automorphism of the Lie algebra . We denote this map by Ad"g"::To say that Ad"g" is a Lie algebra automorphism is to say that Ad"g" is alinear transformation of that preserves theLie bracket . The
which sends "g" to Ad"g" is called the adjoint representation of "G". This is indeed a representation of "G" since is aLie subgroup of and the above adjoint map is a Lie group homomorphism. The dimension of the adjoint representation is the same as the dimension of the group "G".Adjoint representation of a Lie algebra
One may always pass from a representation of a Lie group "G" to a representation of its Lie algebra by taking the derivative at the identity. Taking the derivative of the adjoint
gives the adjoint representation of the Lie algebra ::Here is the Lie algebra of which may be identified with the derivation algebra of . The adjoint representation of a Lie algebra is related in a fundamental way to the structure of that algebra. In particular, one can show that:for all . For more information see: "adjoint representation of a Lie algebra ".Examples
*If "G" is abelian of dimension "n", the adjoint representation of "G" is the trivial "n"-dimensional representation.
*If "G" is amatrix Lie group (i.e. a closed subgroup of "GL"(n,C)), then its Lie algebra is an algebra of "n"×"n" matrices with the commutator for a Lie bracket (i.e. a subalgebra of ). In this case, the adjoint map is given by Ad"g"("x") = "gxg"−1.
*If "G" is SL2(R) (real 2×2 matrices withdeterminant 1), the Lie algebra of "G" consists of real 2×2 matrices with trace 0. The representation is equivalent to that given by the action of "G" by linear substitution on the space of binary (i.e., 2 variable)quadratic form s.Properties
The following table summarizes the properties of the various maps mentioned in the definition
The image of "G" under the adjoint representation is denoted by Ad"G". If "G" is connected, the kernel of the adjoint representation coincides with the kernel of Ψ which is just the center of "G". Therefore the adjoint representation of a connected Lie group "G" is faithful if and only if "G" is centerless. More generally, if "G" is not connected, then the kernel of the adjoint map is the
centralizer of theidentity component "G"0 of "G". By thefirst isomorphism theorem we have:Roots of a semisimple Lie group
If "G" is semisimple, the non-zero weights of the adjoint representation form a
root system . To see how this works, consider the case "G"=SL"n"(R).We can take the group of diagonal matrices diag("t"1,...,"t""n") as ourmaximal torus "T". Conjugation by an element of "T" sends:
Thus, "T" acts trivially on the diagonal part of the Lie algebra of "G" and with eigenvectors "t""i""t""j"-1 on the various off-diagonal entries. The roots of "G" are the weightsdiag("t"1,...,"t""n")→"t""i""t""j"-1. This accounts for the standard description of the root system of "G"=SL"n"(R) as the set of vectors of the form "e""i"−"e""j".
Variants and analogues
The adjoint representation can also be defined for
algebraic group s over any field.The co-adjoint representation is the
contragredient representation of the adjoint representation.Alexandre Kirillov observed that the orbit of any vector in a co-adjoint representation is asymplectic manifold . According to the philosophy inrepresentation theory known as the orbit method (see also theKirillov character formula ), the irreducible representations of a Lie group "G" should be indexed in some way by its co-adjoint orbits. This relationship is closest in the case ofnilpotent Lie group s.References
*Fulton-Harris
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