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:

    “If the national security is involved, anything goes. There are no rules. There are people so lacking in roots about what is proper and what is improper that they don’t know there’s anything wrong in breaking into the headquarters of the opposition party.”
    —Helen Gahagan Douglas (1900–1980)

    “The flower-fed buffaloes of the spring
    In the days of long ago,
    Ranged where the locomotives sing
    And the prairie flowers lie low:—”
    —Vachel Lindsay (1879–1931)

    “The boys think they can all be athletes, and the girls think they can all be singers. That’s the way to fame and success. ...as a group blacks must give up their illusions.”
    —Kristin Hunter (b. 1931)