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The Universal Enveloping Algebra

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Part of the Encyclopaedia of Mathematical Sciences book series (EMS,volume 20)


As is well known (see §19 of Encycl. Math. Sc. 11) every associative algebra A can be turned into a Lie algebra L(A) by replacing its multiplication (a, b) → ab by the commutator [a, b] = ab — ba. Clearly, every homomorphism of associative algebras is automatically a homomorphism of the corresponding Lie algebras, i.e. we have a functor L from the category of associative algebras to the category of Lie algebras. In this chapter we shall consider a functor U, which acts in the opposite direction. In this case a Lie algebra g is embedded into the corresponding associative algebra U(g) as a subalgebra (with respect to the commutator) and generates U(g) as an associative algebra. The algebra U(g) is called the universal enveloping algebra of the Lie algebra g. It was first considered in the year 1899 by Poincaré, who introduced it as a certain algebra of differential operators on the corresponding Lie group (see 2.2 below). The universal enveloping algebra makes it possible to look from a different viewpoint at the Lie functor considered in Chap. 2. In particular, in this way one proves the equivalence of the categories of local analytic Lie groups and finite-dimensional Lie algebras. An important role is played by the Campbell-Hausdorff formula (see §3).


  • Vector Space
  • Associative Algebra
  • Formal Power Series
  • Algebra Homomorphism
  • Universal Envelop Algebra

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© 1993 Springer-Verlag Berlin Heidelberg

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Onishchik, A.L. (1993). The Universal Enveloping Algebra. In: Onishchik, A.L. (eds) Lie Groups and Lie Algebras I. Encyclopaedia of Mathematical Sciences, vol 20. Springer, Berlin, Heidelberg.

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