adjoint representation
\DeclareMathOperator\ad
ad \DeclareMathOperator\EndEnd
Let be a Lie algebra. For every we define the , a.k.a. the adjoint action,
to be the linear transformation with action
For any vector space , we use to denote the Lie algebra of determined by the commutator bracket. So as vector spaces, only the multiplications are different.
In this notation, treating as a vector space, the linear mapping with action
is called the adjoint representation of . The fact that defines a representation is a straight-forward consequence of the Jacobi identity axiom. Indeed, let be given. We wish to show that
where the bracket on the left is the multiplication structure, and the bracket on the right is the commutator bracket. For all the left hand side maps to
while the right hand side maps to
Taking skew-symmetry of the bracket as a given, the equality of these two expressions is logically equivalent to the Jacobi identity:
Title | adjoint representation |
---|---|
Canonical name | AdjointRepresentation |
Date of creation | 2015-10-05 17:38:19 |
Last modified on | 2015-10-05 17:38:19 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 9 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 17B10 |
Related topic | IsotropyRepresentation |
Defines | adjoint action |
Defines | gl |
Defines | general linear Lie algebra |