ADJOINT REPRESENTATION

In mathematics, the 'adjoint representation' (or 'adjoint action') of a Lie group ''G'' is the natural representation of ''G'' on its own Lie algebra. This representation is the linearized version of the action of ''G'' on itself by conjugation.

Contents

Formal definition

Adjoint representation of a Lie algebra

Examples

Properties

Roots of a semisimple Lie group

Variants and analogues

References

Formal definition

Let ''G'' be a Lie group and let $mathfrak\; g$ be its Lie algebra (which we identify with ''T''_''e''''G'', the tangent space to the identity element in ''G''). Define a map
:$Psi\; :\; G\; o\; mathrm\{Aut\}(G),$
by the equation $Psi(g)=\; Psi\_g$ for all ''g'' in ''G'', where the automorphism $Psi\_g$ is defined by
:$Psi\_g(h)\; =\; ghg^\{-1\},$
for all ''h'' in ''G''. It follows that the derivative of Ψ_''g'' at the identity is an automorphism of the Lie algebra $mathfrak\; g$. We denote this map by Ad_''g'':
:$mathrm\{Ad\}\_gcolon\; mathfrak\; g\; o\; mathfrak\; g.$
To say that Ad_''g'' is a Lie algebra automorphism is to say that Ad_''g'' is a linear transformation of $mathfrak\; g$ that preserves the Lie bracket. The map
:$mathrm\{Ad\}colon\; G\; o\; mathrm\{Aut\}(mathfrak\; g)$
which sends ''g'' to Ad_''g'' is called the 'adjoint representation' of ''G''. This is indeed a representation of ''G'' since $mathrm\{Aut\}(mathfrak\; g)$ is a Lie subgroup of $mathrm\{GL\}(mathfrak\; g)$ 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 map
:$mathrm\{Ad\}colon\; G\; o\; mathrm\{Aut\}(mathfrak\; g)$
gives the 'adjoint representation' of the Lie algebra $mathfrak\; g$:
:$mathrm\{ad\}colon\; mathfrak\; g\; o\; mathrm\{Der\}(mathfrak\; g).$
Here $mathrm\{Der\}(mathfrak\; g)$ is the Lie algebra of $mathrm\{Aut\}(mathfrak\; g)$ which may be identified with the derivation algebra of $mathfrak\; g$. 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
:$mathrm\{ad\}\_x(y)\; =\; [x,y],$
for all $x,y\; in\; mathfrak\; g$. 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 a matrix 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 $mathfrak\{gl\}\_n(mathbb\; C)$). In this case, the adjoint map is given by Ad_''g''(''x'') = ''gxg''⁻¹.

★ If ''G'' is SL₂('R') (real 2×2 matrices with determinant 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 forms.

Properties

The following table summarizes the properties of the various maps mentioned in the definition
{| align=center border=1 cellpadding=2 style="border: solid 1pt black; border-collapse: collapse;"
| align=center | $Psicolon\; G\; o\; mathrm\{Aut\}(G),$
| align=center | $Psi\_gcolon\; G\; o\; G,$
|-
| valign=top | Lie group homomorphism:

• $Psi\_\{gh\}\; =\; Psi\_gPsi\_h$

| valign=top | Lie group automorphism:

• $Psi\_g(ab)\; =\; Psi\_g(a)Psi\_g(b)$


• $(Psi\_g)^\{-1\}\; =\; Psi\_\{g^\{-1\}\}$

|-
| align=center | $mathrm\{Ad\}colon\; G\; o\; mathrm\{Aut\}(mathfrak\; g)$
| align=center | $mathrm\{Ad\}\_gcolon\; mathfrak\; g\; o\; mathfrak\; g$
|-
| valign=top | Lie group homomorphism:

• $mathrm\{Ad\}\_\{gh\}\; =\; mathrm\{Ad\}\_gmathrm\{Ad\}\_h$

| valign=top | Lie algebra automorphism:

• $mathrm\{Ad\}\_g$ is linear
  
• $(mathrm\{Ad\}\_g)^\{-1\}\; =\; mathrm\{Ad\}\_\{g^\{-1\}\}$
  
• $mathrm\{Ad\}\_g[x,y]\; =\; [mathrm\{Ad\}\_g(x),mathrm\{Ad\}\_g(y)]$
  

|-
| align=center | $mathrm\{ad\}colon\; mathfrak\; g\; o\; mathrm\{Der\}(mathfrak\; g)$
| align=center | $mathrm\{ad\}\_xcolon\; mathfrak\; g\; o\; mathfrak\; g$
|-
| valign=top | Lie algebra homomorphism:

• $mathrm\{ad\}$ is linear


• $mathrm\{ad\}\_\{[x,y]\}\; =\; [mathrm\{ad\}\_x,mathrm\{ad\}\_y]$

| valign=top | Lie algebra derivation:

• $mathrm\{ad\}\_x$ is linear
  
• $mathrm\{ad\}\_x[y,z]\; =\; [mathrm\{ad\}\_x(y),z]\; +\; [y,mathrm\{ad\}\_x(z)]$
  

|}
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 the identity component ''G''₀ of ''G''. By the first isomorphism theorem we have
:$mathrm\{Ad\}\_G\; cong\; G/C\_G(G\_0).$

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 our maximal torus ''T''. Conjugation by an element of ''T'' sends
:
a_{11}&a_{12}&cdots&a_{1n}\
a_{21}&a_{22}&cdots&a_{2n}\
dots&dots&ddots&dots\
a_{n1}&a_{n2}&cdots&a_{nn}\
end{bmatrix}
mapsto
egin{bmatrix}
a_{11}&t_1t_2^{-1}a_{12}&cdots&t_1t_n^{-1}a_{1n}\
t_2t_1^{-1}a_{21}&a_{22}&cdots&t_2t_n^{-1}a_{2n}\
dots&dots&ddots&dots\
t_nt_1^{-1}a_{n1}&t_nt_2^{-1}a_{n2}&cdots&a_{nn}\
end{bmatrix}.

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 weights
diag(''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 groups 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 a symplectic manifold. According to the philosophy in representation theory known as the 'orbit method' (see also the Kirillov 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 of nilpotent Lie groups.

References

★ Representation Theory: A First Course, , William, Fulton, Springer, 1991,