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liealgebrarepresentation

lie algebra representation

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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...

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Lie algebra representation description
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.

Lie algebra representation description

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.

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