# nLab Poisson n-algebra

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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 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.

This fact is a higher analog of Kontsevich formality. It means that every higher dimensional prequantum field theory giveny 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.

### 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

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.

For general discusison of the relation to E-n algebras see

Revised on August 4, 2015 14:01:14 by Urs Schreiber (147.231.89.193)