nLab PBW theorem




Lie theory

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\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



Given a Lie algebra gg which is flat as a kk-module over a commmutative ground ring kk\supset\mathbb{Q} containing the rationals, consider the universal enveloping algebra U(g)= i=0 U i(g)U(g)=\cup_{i= 0}^\infty U^i(g) as a filtered algebra and a Hopf algebra, where gg is the subspace of primitive elements.

The Poincaré–Birkhoff–Witt theorem (often abbreviated to PBW theorem) says that the associated graded algebra is canonically isomorphic to the symmetric algebra Sym(g)Sym(g) as an algebra, and the projection U(g)GrU(g)Sym(g)U(g)\to Gr U(g)\cong Sym(g) is an isomorphism of kk-coalgebras.

The fact that associated graded algebra is isomorphic to the symmetric algebra is a weak form of the theorem and it is usually proved either by very explicit and long calculation constructing certain representation or by application of the diamond lemma. The induced Poisson structure on Sym(g)Sym(g) is the linear Poisson structure for the corresponding gg.

The notions of Lie algebra, symmetric algebra, enveloping algebra, etc. can also be formulated in the context of a kk-linear tensor category (with finite direct sums and split idempotents, and also with countable coproducts over which the tensor product distributes in case one wants to work with ungraded objects), and the PBW theorem remains valid in that context. In particular, the PBW theorem may be formulated and proven for super Lie algebras. Details may be found in Deligne-Morgan.

Abstract framework

An abstract framework for the Poincar'e-Birkhoff-Witt theorem in the setting of natural transformations of monads was developed by Dotsenko and Tamaroff.


  • Pierre-Paul Grivel, Une histoire du théorème de Poincaré-Birkhoff-Witt, Expo. Math. 22 (2004), no. 2, 145–184 MR2005b:17024 doi

  • Vladimir Dotsenko, Pedro Tamaroff, Endofunctors and Poincaré-Birkhoff-Witt theorems, arXiv:1804.06485.

Last revised on December 27, 2019 at 21:15:33. See the history of this page for a list of all contributions to it.