Enveloping algebra

In Mathematical, one can build the enveloping algebra U (L) of a Algèbre of Dregs L . It acts a unit algebra associative which makes it possible to give an account of the majority of the properties of L .

If has is an associative algebra on a body K , one can easily provide it with a structure of algebra of Dregs, by posing =xy-yx . One notes the algebra of Dregs thus obtained A_L .

The construction of an enveloping algebra answers the reciprocal problem: starting from an algebra of Dregs, one builds an associative algebra whose switch corresponds to the hook which one had left.

Construction

That is to say L an algebra of Dregs on a body K . That is to say T (L) the tensorial Algebra of L . One builds U (L) there starting from T (L) by imposing the relations

x \ otimes there \ otimes x=

More formally, one notes I the Idéal bilatère generated by the $x \ otimes there there \ otimes x-$ . U (L) is then the quotient of T (L) by the ideal I . The canonical injection of L in T (L) provides then a morphism $\ iota: L \ to U (L)$ .

Universal property

One can characterize the enveloping algebra of L by the following universal property: U (L) is the single assocative algebra such as for all associative K-algebra has and any morphism of algebra of Dregs $\ phi: L \ to A_L$ , there exists a single morphism of associative algebra $\ Phi: U (L) \ to A$ such as $\ phi= \ Phi \ circ \ iota$ .

Other properties

the interest first of the construction of the enveloping algebra is that any representation of an algebra of Dregs L can be seen like a module on U (L) . Formally, there is a equivalence of categories between the representations of L and the U (L) - modules.

the Théorème of Poincaré-Birkhoff-Witt makes it possible to better include/understand the structure of the enveloping algebra. An important corollary of this theorem is that the above definite application $\ iota$ is injective.

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