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 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

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