nLab universal enveloping E-n algebra

Idea

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

$$E_n Alg \stackrel{\overset{\mathcal{U}_n}{\leftarrow}}{\underset{}{\to}} L_\infty$$

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

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

Examples and applications

• Weyl n-algebra

References

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

• Tom Lada, Martin Markl, Strongly homotopy Lie algebras (arXiv:hep-th/9406095)

and more details have been worked out here:

• Vladimir Baranovsky, A universal enveloping for L-infinity algebras (https://arxiv.org/abs/0706.13960)

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

• David Ayala, John Francis, Factorization homology of topological manifolds (arXiv:1206.5522)

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

See also

• John Francis, Dennis Gaitsgory, Chiral Koszul duality (arXiv:1103.5803)