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Concept
Lie algebra representation
Summary
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
The notion is closely related to that of a representation of a Lie group. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.
In the study of representations of a Lie algebra, a particular ring, called the universal enveloping algebra, associated with the Lie algebra plays an important role. The universality of this ring says that the of representations of a Lie algebra is the same as the category of modules over its enveloping algebra.
Let be a Lie algebra and let be a vector space. We let denote the space of endomorphisms of , that is, the space of all linear maps of to itself. We make into a Lie algebra with bracket given by the commutator: for all ρ,σ in . Then a representation of on is a Lie algebra homomorphism
Explicitly, this means that should be a linear map and it should satisfy
for all X, Y in . The vector space V, together with the representation ρ, is called a -module. (Many authors abuse terminology and refer to V itself as the representation).
The representation is said to be faithful if it is injective.
One can equivalently define a -module as a vector space V together with a bilinear map such that
for all X,Y in and v in V. This is related to the previous definition by setting X ⋅ v = ρ(X)(v).
Adjoint representation of a Lie algebra
The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra on itself:
Indeed, by virtue of the Jacobi identity, is a Lie algebra homomorphism.
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