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.

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 a decisive 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.

A representation of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra homomorphism

from ${\displaystyle {\mathfrak {g}}}$ to the Lie algebra of endomorphisms on a vector space V (with the commutator as the Lie bracket), sending an element x of ${\displaystyle {\mathfrak {g}}}$ to an element ρ_x of ${\displaystyle {\mathfrak {gl}}(V)}$.

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