symmetric monoidal (∞,1)-category of spectra
For $n \in \mathbb{N}$, a Poisson $n$-algebra $A$ is a Poisson algebra $A$ in a category of chain complexes with Poisson bracket of degree $(1-n)$ (which is a bracket of degree 0 on $\mathbf{B}^{n-1} A$).
The homology of an E-n algebra for $n \geq 2$ is a Poisson $n$-algebra.
Moreover, in chain complexes over a field of characteristic zero the E-n operad is formal, hence equivalent to its homology, and so in this context $E_n$-algebras are equivalent to Poisson $n$-algebras.
This fact is a higher analog of Kontsevich formality. It means that every higher dimensional prequantum field theory given by a $P_n$ algebra does have a deformation quantization (as factorization algebras) and that the space of choice of these a torsor over the automorphism infinity-group of $E_n$, a higher analog of the Grothendieck-Teichmüller group.
See also tho MO discussion linked to below.
There is a forgetful functor from Poisson $n$-algebras to dg-Lie algebras given by forgetting the associative algebra structure and by shifting the underlying chain complex by $(n-1)$.
Conversely, this functor has a derived left adjoint which sends a dg-Lie algebra $(\mathfrak{g},d)$ to its universal enveloping Poisson n-algebra $(Sym(\mathfrak{g}[n-1], d))$. (See also Gwilliam, section 4.5).
A Poisson 1-algebra is a Poisson algebra.
A Poisson 2-algebra is a Gerstenhaber algebra.
The Chevalley-Eilenberg algebra of a symplectic Lie n-algebroid (see there for details) is naturally a Poisson $(1+n)$-algebra.
A classical BV complex is naturally (if obtained as a derived critical locus, or else by definition) a Poisson 0-algebra.
duality between algebra and geometry in physics:
algebraic deformation quantization
dimension | classical field theory | Lagrangian BV quantum field theory | factorization algebra of observables |
---|---|---|---|
general $n$ | P-n algebra | BD-n algebra? | E-n algebra |
$n = 0$ | Poisson 0-algebra | BD-0 algebra? = BD algebra | E-0 algebra? = pointed space |
$n = 1$ | P-1 algebra = Poisson algebra | BD-1 algebra? | E-1 algebra? = A-∞ algebra |
An introduction to Poisson $n$-algebras in dg-geometry/symplectic Lie n-algebroids is in section 4.2 of
For discussion in the context of perturbative quantum field theory/factorization algebras/BV-quantization see
Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory : $P_0$-operad (wiki, pdf)
Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)
and for further references along these lines see at factorization algebra.
For general discusison of the relation to E-n algebras see