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

Properties

Relation to $E_n$-algebras

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 (see the little n-disk 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 choices of these is 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.

Relation to $L_\infty$-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 $(\mathfrak{g},d)$ to its universal enveloping Poisson n-algebra $(Sym(\mathfrak{g}[n-1], d))$. (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

• BV-algebra

• higher Poisson structure

  • Nambu bracket

  • Poisson bracket Lie n-algebra

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

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

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 (wikilass=‘newWikiWord’>P_0%20operad?</span>), 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

• MO discussion, Poisson algebras as deformations vs. Poisson algebras in algebraic topology