Lie algebra representation

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 V {displaystyle V} together with a collection of operators on V {displaystyle V} satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators. 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 V {displaystyle V} together with a collection of operators on V {displaystyle V} 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 category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra. Let g {displaystyle {mathfrak {g}}} be a Lie algebra and let V {displaystyle V} be a vector space. We let g l ( V ) {displaystyle {mathfrak {gl}}(V)} denote the space of endomorphisms of V {displaystyle V} , that is, the space of all linear maps of V {displaystyle V} to itself. We make g l ( V ) {displaystyle {mathfrak {gl}}(V)} into a Lie algebra with bracket given by the commutator: [ X , Y ] = X Y − Y X {displaystyle =XY-YX} . Then a representation of g {displaystyle {mathfrak {g}}} on V {displaystyle V} is a Lie algebra homomorphism Explicitly, this means that ρ {displaystyle ho } should be a linear map and it should satisfy for all X {displaystyle X} and Y {displaystyle Y} in g {displaystyle {mathfrak {g}}} . The vector space V, together with the representation ρ, is called a g {displaystyle {mathfrak {g}}} -module. (Many authors abuse terminology and refer to V itself as the representation). The representation ρ {displaystyle ho } is said to be faithful if it is injective. One can equivalently define a g {displaystyle {mathfrak {g}}} -module as a vector space V together with a bilinear map g × V → V {displaystyle {mathfrak {g}} imes V o V} such that for all x,y in g {displaystyle {mathfrak {g}}} and v in V. This is related to the previous definition by setting x ⋅ v = ρx (v).