Adjoint Representation of A Lie Group - Examples

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 ). In this case, the adjoint map is given by Adg(x) = gxg−1.
  • If G is SL(2, 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.

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

Famous quotes containing the word examples:

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
    —Bernard Mandeville (1670–1733)