# nLab Beilinson-Drinfeld algebra

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

A Beilinson-Drinfeld algebra (or BD-algebra for short) is like a BV-algebra with nilpotent BV-operator, but over formal power series in one formal parameter $\hbar$, and such that the Gerstenhaber bracket is proportional to that parameter.

This means that one may think of a BD-algebra as an $\hbar$-parameterized formal family of algebras which for $\hbar = 0$ are Poisson 0-algebras and for $\hbar \neq 0$ they look superficially like BV-algebras. But see remark below.

Such BD-algbras are used to formalize formal deformation quantization in the context of the BV-BRST formalism (in Costello-Gwilliam).

## Definition

###### Definition

A quantum BV complex or Beilinson-Drinfeld algebra is

1. a differential graded-commutative algebra $A$ (whose differential we denote by $\Delta$) over the ring $\mathbb{R} [ [ \hbar ] ]$ of formal power series over the real numbers in a formal constant $\hbar$,

2. equipped with a Poisson bracket $\{-,-\}$ of the same degree as the differential

such that

• the following equation holds for all elements $a,b \in A$ of homogeneous degree ${\vert a\vert}, {\vert b\vert} \in \mathbb{Z}$

$\Delta( a \cdot b) = (\Delta a) \cdot b + (-1)^{\vert a\vert} a \Delta b + \hbar \{a,b\}$
###### Remark

If in the equation in def. one replaces $\hbar$ by 1, then it takes the form characteristic of a BV-algebra. However the differential $\Delta$ here is the differential in the underlying chain complex for an algebra over an operad in chain complexes, while in a BV-algebra $\Delta$ is the unary operation encoded in the BV-operad, hence present already for algebras in plain modules/chain complexes over that operad. Accordingly, in the context of BD-algebra then for $\hbar \neq 0$ the bracket is actually trivial up to homotopy.

For more see at relation between BV and BD.

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

The notion was introduced in

A discussion is in section 2.4 of