nLab
Poisson n-algebra

Context

Higher algebra

Symplectic geometry

Contents

Definition

For nn \in \mathbb{N}, a Poisson nn-algebra AA is a Poisson algebra AA in a category of chain complexes with Poisson bracket of degree (1n)(1-n) (which is a bracket of degree 0 on B n1A\mathbf{B}^{n-1} A).

Properties

Relation to E nE_n-algebras

The homology of an E-n algebra for n2n \geq 2 is a Poisson nn-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 nE_n-algebras are equivalent to Poisson nn-algebras.

This fact is a higher analog of Kontsevich formality. It means that every higher dimensional prequantum field theory giveny by a P nP_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 nE_n, a higher analog of the Grothendieck-Teichmüller group.

See also tho MO discussion linked to below.

Relation to L L_\infty-algebras

There is a forgetful functor from Poisson nn-algebras to dg-Lie algebras given by forgetting the associative algebra structure and by shifting the underlying chain complex by (n1)(n-1).

Conversely, this functor has a derived left adjoint which sends a dg-Lie algebra (𝔤,d)(\mathfrak{g},d) to its universal enveloping Poisson n-algebra (Sym(𝔤[n1],d))(Sym(\mathfrak{g}[n-1], d)). (See also Gwilliam, section 4.5).

Examples

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

algebraic deformation quantization

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

Revised on November 7, 2013 03:52:57 by Urs Schreiber (77.251.114.72)