Any Lie group is a representation of itself (via ) and the tangent space is mapped to itself by the group action. This gives the linear adjoint representation.
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2 Variants and analogues 3 Roots of a semisimple Lie group |
The adjoint representation 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 contragredient representation of the adjoint representation. A. 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, the irreducible representations of a Lie group G should be indexed in some way by its co-adjoint orbits.
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=SLn(R).
We can take the group of diagonal matrices diag(t1,...,tn) as our maximal 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 titj-1 on the various off-diagonal entries. The roots of G are the weights
diag(t1,...,tn)→titj-1. This accounts for the standard desciption of the root system of G=SLn(R) as the set of vectors of the form ei-ej.Examples
Variants and analogues
Roots of a semisimple Lie group