Editing Lie algebra representation (section)

{{Short description|Writing Lie algebra sets as matrices}}
{{Lie groups |Representation}}

In the [[mathematics|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 [[matrix (mathematics)|matrices]] (or [[endomorphism]]s 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 <math>V</math> together with a collection of operators on <math>V</math> satisfying some fixed set of commutation relations, such as the relations satisfied by the [[angular momentum operator]]s.

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 (mathematics)|ring]], called the [[universal enveloping algebra]], associated with the Lie algebra plays an important role. The universality of this ring says that the [[category (mathematics)|category]] of representations of a Lie algebra is the same as the category of [[module (mathematics)|module]]s over its enveloping algebra.<!--(This is very similar to the case of [[group ring]].) Furthermore, since the center ''Z'' of the enveloping algebra is a commutative ring and it acts on Lie algebra representations, Lie algebra representations may be thought of as sheaves on the [[spectrum of a ring|spectrum]] of ''Z''. In the recent developments, this appralch has been exploited extensively, making the subject largely a part of [[algebraic geometry]].-->