# nLab Poisson n-algebra

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

For $n\in ℕ$, a Poisson $n$-algebra $A$ is a Poisson algebra $A$ in a category of chain complexes with Poisson bracket of degree $\left(1-n\right)$ (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 ${L}_{\infty }$-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 $\left(n-1\right)$.

Conversely, this functor has a derived left adjoint which sends a dg-Lie algebra $\left(𝔤,d\right)$ to its universal enveloping Poisson n-algebra $\left(\mathrm{Sym}\left(𝔤\left[n-1\right],d\right)\right)$. (See also Gwilliam, section 4.5).

## Examples

duality between algebra and geometry in physics:

algebraic deformation quantization

dimensionclassical field theoryLagrangian BV quantum field theoryfactorization algebra of observables
general $n$P-n algebraBD-n algebra?E-n algebra
$n=0$Poisson 0-algebraBD-0 algebra? = BD algebraE-0 algebra? = pointed space
$n=1$P-1 algebra = Poisson algebraBD-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

For discussion in the context of perturbative quantum field theory/factorization algebras/BV-quantization see

and for further references along these lines see at factorization algebra.

Revised on April 22, 2013 15:24:33 by Urs Schreiber (89.204.139.130)