nLab
universal enveloping E-n algebra

Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Higher algebra

Contents

Idea

For all nn \in \mathbb{N} there is supposed to be a pair of adjoint (∞,1)-functors

E nAlg𝒰 nL E_n Alg \stackrel{\overset{\mathcal{U}_n}{\leftarrow}}{\underset{}{\to}} L_\infty

between E-n algebras and L-∞ algebras, suitably factoring through Poisson n-algebras.

The left adjoint 𝒰 n\mathcal{U}_n sends an L-∞ algebra to its universal enveloping E nE_n-algebra in that for n=1n = 1 and for 𝔤\mathfrak{g} an ordinary Lie algebra, 𝒰 1(𝔤)\mathcal{U}_1(\mathfrak{g}) is the associative algebra (an E 1E_1=A-∞ algebra) which is the ordinary universal enveloping algebra of 𝔤\mathfrak{g}.

Examples and applications

References

Discussion for n=1n = 1, hence universal A-∞-enveloping algebras of L-∞ algebras is around theorem 3.1, 3.3 in

and more details have been worked out here:

Aspects of general enveloping E nE_n-alebras are mentioned in the context of factorization homology in section 5 and in particular around the bottom of p. 18 in

and more specifically in the context of factorization algebras of observables around remark 4.5.5 of

  • Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)

The fact that 𝒰 1\mathcal{U}_1 reproduces the traditional universal enveloping algebra of a Lie algebra is prop. 4.6.1 in (Gwilliam).

See also

Last revised on August 30, 2019 at 16:13:27. See the history of this page for a list of all contributions to it.