Adjoint representation

The adjoint representation is the linear action of a Lie group on its Lie algebra induced by conjugation. It describes how group elements transform infinitesimal generators.

Group conjugation: The operation sending an element g and an element h of a group to ghg⁻¹. Conjugating group elements is the operation whose derivative defines the adjoint action.
Tangent space at the identity: The tangent space of a Lie group at its identity element, identified with its Lie algebra. It is the space on which differentiated conjugation acts.
General linear group: The group of invertible linear transformations of a finite-dimensional vector space. For this group, the adjoint action is matrix conjugation on its Lie algebra.
Co-adjoint representation: The action of a Lie group on the dual of its Lie algebra induced by the adjoint action. It acts on linear functionals rather than on Lie algebra elements.
Lie algebra: A vector space with a bilinear bracket satisfying antisymmetry and the Jacobi identity. The adjoint representation acts on this vector space.
Group representation: A homomorphism from a group into the invertible linear maps of a vector space. The adjoint representation is a canonical representation built from the group itself.
Special linear group: The group of matrices with determinant one over a specified field. Its adjoint action on trace-zero matrices is a standard concrete example.
Adjoint representation of a Lie algebra: The representation sending each Lie algebra element to the operation of bracketing with it. It is the infinitesimal representation, distinct from the group-level action.
Differential of a map: The linear map induced between tangent spaces by a smooth map between manifolds. Differentiating conjugation at the identity yields the group’s adjoint representation.
Lie group homomorphism: A smooth group homomorphism between Lie groups. Conjugation and its derivative are smooth maps compatible with the group structure.

Linked from 6 pages