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In mathematics, for any Lie algebra one can construct its universal enveloping algebra . This construction passes from the non-associative structure to a (more familiar, and possibly easier to handle) unital associative algebra which captures the important properties of . Any associative algebra over the field becomes a Lie algebra over with the Lie bracket: :. That is, from an associative product, one can construct a Lie bracket by taking the commutator with respect to that associative product. Denote this Lie algebra by . Construction of the universal enveloping algebra attempts to reverse this process: to a given Lie algebra over , find the "most general" unital associative -algebra such that the Lie algebra contains ; this algebra is . The important constraint is to preserve the representation theory: the representations of correspond in a one-to-one manner to the modules over . In a typical context where is acting by ''infinitesimal transformations'', the elements of act like differential operators, of all orders. Next to Lie algebras, the construction of the universal enveloping algebra has been generalized for Malcev algebras,〔J.M. Perez-Izquierdo, I.P. Shestakov: ''An envelope for Malcev algebras'', Journal of Algebra 272 (2004) 379–393.〕 Bol algebras 〔J.M. Perez-Izquierdo: ''An envelope for Bol algebras'', Journal of Algebra 284 (2005) 480–493.〕 and left alternative algebras.〔Rukavicka Josef: ''An envelope for left alternative algebras'', International Journal of Algebra, Vol. 7, 2013, no. 10, 455–462, ()〕 ==Motivation== An important topic in Lie algebras studies and probably the main source of their appearance in applications is representation of the Lie algebra. A representation assigns to any element of a Lie algebra a linear operator . The space of linear operators is not only a Lie algebra, but also an associative algebra and so one can consider products . The main point to introduce the universal enveloping algebra is to study such products in various representations of a Lie algebra. One obstacle can be immediately seen in a naive attempt to do this: properties of products drastically depend on the representation, not only on the Lie algebra itself. For example for one representation we might have , while in another representation this product may not be zero. Nevertheless it appears to be true that certain properties are universal for all representations, i.e. they hold true for all representations simultaneously. The universal enveloping algebra is a way to grasp all such properties and only them. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Universal enveloping algebra」の詳細全文を読む スポンサード リンク
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