Adjoint representation

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In mathematics , the adjoint representations of Lie groups and Lie algebras play an important role in differential geometry , representation theory and mathematical physics.

Lie groups and Lie algebras

A Lie group is a differentiable manifold , which also has the structure of a group , so that the group connection and the inversion can be differentiated as often as desired .

The Lie algebra of a Lie group is the vector space of the left-invariant vector fields with the commutator as Lie bracket . The Lie algebra can be canonically identified with the tangent space in the neutral element of the Lie group :

.

Adjoint representations

Let be a Lie group with Lie algebra .

The conjugation with an element is through

defined figure .

The adjoint representation

is through

for all defined Lie group homomorphism .

The induced Lie algebra homomorphism is also referred to as an adjoint representation

.

Because, according to Lie's theorems, for every finite-dimensional real Lie algebra there is a unambiguous, simply connected Lie group with the exception of isomorphism , the adjoint representation can be defined for every such Lie algebra.

Explicit description

The adjoint representation of a Lie algebra corresponds to using the Lie bracket: it applies

for everyone .

For matrix groups , i. H. closed subgroups of , the adjoint representation of the Lie group can also be explicitly described: after the canonical identification of with a subset of holds

for everyone .

literature

  • Arvanitoyeorgos, Andreas: An introduction to Lie groups and the geometry of homogeneous spaces. Translated from the 1999 Greek original and revised by the author. Student Mathematical Library, 22nd American Mathematical Society, Providence, RI, 2003. ISBN 0-8218-2778-2
  • Hall, Brian C .: Lie groups, Lie algebras, and representations. An elementary introduction. Graduate Texts in Mathematics, 222. Springer-Verlag, New York, 2003. ISBN 0-387-40122-9
  • Knapp, Anthony W .: Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhauser Boston, Inc., Boston, MA, 2002. ISBN 0-8176-4259-5