Adjoint Representation of A Lie Group - Roots of A Semisimple Lie Group

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(t1, ..., tn) 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 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 description of the root system of G = SLn(R) as the set of vectors of the form ei−ej.

Read more about this topic:  Adjoint Representation Of A Lie Group

Famous quotes containing the words roots, lie and/or group:

    “Einstein is not ... merely an artist in his moments of leisure and play, as a great statesman may play golf or a great soldier grow orchids. He retains the same attitude in the whole of his work. He traces science to its roots in emotion, which is exactly where art is also rooted.”
    —Havelock Ellis (1859–1939)

    “Even as fog continues to lie in the valleys, so does ancient sin cling to the low places, the depressions in the world consciousness.”
    —Dewitt Bodeen (1908–1988)

    “We begin with friendships, and all our youth is a reconnoitering and recruiting of the holy fraternity they shall combine for the salvation of men. But so the remoter stars seem a nebula of united light, yet there is no group which a telescope will not resolve; and the dearest friends are separated by impassable gulfs.”
    —Ralph Waldo Emerson (1803–1882)