Adjoint representation

The adjoint representation of a Lie group ''G'' is the linearized version of the action of ''G'' on itself by conjugation. For each ''g'' in ''G'', the inner automorphism ''x''→''gxg''^-1 gives a linear transformation adjoint endomorphism from the Lie algebra of ''G'', i.e., the tangent space of ''G'' at the identity element, to itself. The map Ad(''g'') is called the adjoint endomorphism; the map ''g''→Ad(''g'') is the adjoint representation. Any Lie group is a Group representation of itself (via

h\rightarrow ghg^{-1}

) and the tangent space is mapped to itself by the group action. This gives the linear adjoint representation. == Examples == *If ''G'' is commutative of dimension ''n'', the adjoint representation of ''G'' is the trivial ''n''-dimensional representation. *The kernel of the adjoint representation of ''G'' is the center of ''G''. *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. == Variants and analogues == The adjoint endomorphism of a Lie algebra ''L'' sends ''x'' in ''L'' to ad(''x''), where :ad(''x'')(y) = [x y]. If ''L'' arises as the Lie algebra of a Lie group ''G'', the usual method of passing from Lie group representations to Lie algebra representations sends the adjoint representation of ''G'' to the adjoint representation of ''L''. The adjoint representation can also be defined for algebraic groups over any field. The co-adjoint representation is the contragradient representation of the adjoint representation. A. Kirillov observed that the Group action#Orbits and stabilizers of any vector in a co-adjoint representation is a symplectic manifold. According to the philosophy in representation theory known as the orbit method, 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. == Roots of a semisimple Lie group == If ''G'' is semisimple group, the non-zero weight (representation theory) 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

\begin{bmatrix}
a_{11}&a_{12}&\cdots&a_{1n}\\
a_{21}&a_{22}&\cdots&a_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
a_{n1}&a_{n2}&\cdots&a_{nn}\\
\end{bmatrix}
\mapsto
\begin{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}\\
\vdots&\vdots&\ddots&\vdots\\
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''. Lie groups Representation theory

Adjoint representation

I really wouldn't say [Cayley's theorem] has much to do with it. There is no obvious relation, for a group G, between acting on itself by conjugation and by translation - very different permutation representations. User:Charles Matthews 06:43, 23 Aug 2003 (UTC) == Any Lie group is a representation of itself? == This does not sound right. "representation" can refer either to a vector space with an action of a group or to the group morphism from G to GL(V). Neither of these applies here. Better: any Lie group G acts on itself...

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