nLab
Poisson n-algebra

Definition

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 $B^{n-1}A$).

Properties

Relation to $E_n$-algebras

The homology of an algebra over an operad over the little n-cubes operad for $n\ge 2$ is a Poisson $n$-algebra.

Moreover, in chain complexes over a field of characteristic 0 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.

Relation to $L_{\mathrm{\infty }}$-algebras

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 $(𝔤,d)$ to its universal enveloping Poisson n-algebra $(\mathrm{Sym}(𝔤[n-1],d))$. (See also Gwilliam, section 4.5).

Examples

• 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.

Related concepts

• Poisson bracket Lie n-algebra

• BV-algebra

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

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

References

• Alberto Cattaneo, Domenico Fiorenza, R. Longoni, Graded Poisson Algebras, Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T. , vol. 2, p. 560-567 (Oxford: Elsevier, 2006). (pdf)

An introduction to Poisson $n$-algebras in dg-geometry/symplectic Lie n-algebroids is in section 4.2 of

• Alberto Cattaneo, Florian Schätz, Introduction to supergeometry, Lecture notes of a course held at the school Poisson 2010 at IMPA, July 2010; 21 (arXiv:1011.3401)

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.