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The derived BV bracket is an odd Poisson bracket

Statement

Let $(A, Δ)$ be as in BV algebra with a degree-one generator. For homogeneous $a, b, c \in A$ , the derived bracket has degree $∣ {a, b} ∣ = ∣ a ∣ + ∣ b ∣ + 1$ and satisfies ${a, b} = - (- 1)^{(∣ a ∣ + 1) (∣ b ∣ + 1)} {b, a},$ ${a, b c} = {a, b} c + (- 1)^{(∣ a ∣ + 1) ∣ b ∣} b {a, c},$ and ${a, {b, c}} = {{a, b}, c} + (- 1)^{(∣ a ∣ + 1) (∣ b ∣ + 1)} {b, {a, c}} .$ Thus it is an odd Poisson bracket, meaning an odd-skew bracket that is a derivation in each argument and satisfies the displayed odd Jacobi identity.

Facts & Assumptions

Given: A degree- $+ 1$ BV generator $Δ$ on a graded-commutative algebra $A$ over a field of characteristic $0$ , and homogeneous $a, b, c \in A$ .

[F1]

BV algebra with a degree-one generator: $Δ (1) = 0$ , $Δ^{2} = 0$ , $[[[Δ, m_{a}], m_{b}], m_{c}] = 0$ , and ${a, b} = (- 1)^{∣ a ∣} (Δ (a b) - (Δ a) b - (- 1)^{∣ a ∣} a Δ b)$ .

Proof

1.1F1algebra

Put $m = ∣ a ∣$ , $n = ∣ b ∣$ , $δ_{a} = [Δ, m_{a}]$ , and $T_{a, b} = [δ_{a}, m_{b}]$ . By [F1], $[T_{a, b}, m_{c}] = 0$ for every homogeneous $c$ , so $T_{a, b} (c) = (- 1)^{∣ T_{a, b} ∣ ∣ c ∣} c T_{a, b} (1) = T_{a, b} (1) c$ ; the two signs cancel by graded commutativity, hence $T_{a, b} = m_{T_{a, b} (1)}$ . Direct evaluation gives $T_{a, b} (1) = Δ (a b) - (- 1)^{m} a Δ b - (- 1)^{(m + 1) n} b Δ a = Δ (a b) - (- 1)^{m} a Δ b - (Δ a) b = (- 1)^{m} {a, b}$ . Thus $[δ_{a}, m_{b}] = m_{(- 1)^{m} {a, b}}$ .

1.2F1algebra

Graded commutativity expands the defining formula to ${b, a} = (- 1)^{n + m n} Δ (a b) - (- 1)^{n + m n + m} a Δ b - (- 1)^{m n + n} (Δ a) b$ . Multiplication by $- (- 1)^{(m + 1) (n + 1)}$ reduces these coefficients, modulo $2$ , to $(- 1)^{m}$ , $- 1$ , and $- (- 1)^{m}$ , respectively, giving exactly ${a, b}$ . Each term of the bracket formula has degree $∣ a ∣ + ∣ b ∣ + 1$ . This proves odd skew-symmetry and the degree assertion.

2.1F1step 1.1algebra

Define $H_{a} = δ_{a} - m_{Δ a}$ . Then $H_{a} (1) = 0$ and $[H_{a}, m_{b}] = m_{(- 1)^{m} {a, b}}$ by step 1.1 because multiplication operators gradedly commute. Evaluating at $1$ gives $H_{a} (b) = (- 1)^{m} {a, b}$ ; evaluating at homogeneous $c$ gives $H_{a} (b c) - (- 1)^{(m + 1) n} b H_{a} (c) = H_{a} (b) c$ . Multiplying by $(- 1)^{m}$ proves ${a, b c} = {a, b} c + (- 1)^{(m + 1) n} b {a, c}$ .

2.2F1step 1.1step 1.2algebra

Expansion of the graded commutators gives $[Δ, [Δ, m_{b}]] = [Δ^{2}, m_{b}] = 0$ and $[δ_{a}, δ_{b}] = [Δ, [m_{a}, δ_{b}]]$ , because the other graded-Jacobi term is a signed multiple of $[m_{a}, [Δ, δ_{b}]] = 0$ . By step 1.1 and odd skew-symmetry from step 1.2, $[m_{a}, δ_{b}] = - (- 1)^{m (n + 1)} [δ_{b}, m_{a}] = - m_{{a, b}}$ , so $[δ_{a}, δ_{b}] = - [Δ, m_{{a, b}}]$ .

2.3F1step 1.1algebra

Rearranging the defining bracket gives $Δ (u v) = (Δ u) v + (- 1)^{∣ u ∣} u Δ v + (- 1)^{∣ u ∣} {u, v}$ . Apply this to $(Δ a) b$ and $a (Δ b)$ when expanding $Δ {a, b}$ ; the two $(Δ a) (Δ b)$ terms cancel and $Δ^{2} = 0$ leaves $Δ {a, b} = {Δ a, b} - (- 1)^{m} {a, Δ b}$ .

3.1step 1.2step 2.1algebra

By steps 1.2 and 2.1, express ${a b, c} = - (- 1)^{(∣ a ∣ + ∣ b ∣ + 1) (∣ c ∣ + 1)} {c, a b}$ , expand ${c, a b}$ by the derivation rule, and use odd skew-symmetry on the two terms; reducing exponents modulo $2$ gives ${a b, c} = a {b, c} + (- 1)^{∣ b ∣ (∣ c ∣ + 1)} {a, c} b$ . Hence the bracket is also a derivation in its first argument and is a biderivation.

4.1F1step 1.1step 1.2step 2.2step 2.3algebra∎

Expanding $[H_{a}, H_{b}]$ using steps 1.1 and 2.2 gives $[H_{a}, H_{b}] = - [Δ, m_{{a, b}}] - (- 1)^{m} m_{{a, Δ b}} + (- 1)^{(m + 1) (n + 1) + n} m_{{b, Δ a}}$ . By step 1.2, ${Δ a, b} = - (- 1)^{m (n + 1)} {b, Δ a}$ , and the last coefficient equals $- (- 1)^{m (n + 1)}$ ; step 2.3 therefore makes the last two terms $m_{Δ {a, b}}$ , so $[H_{a}, H_{b}] = - H_{{a, b}}$ . Thus $D_{a} = {a, -} = (- 1)^{m} H_{a}$ satisfies $[D_{a}, D_{b}] = (- 1)^{m + n + 1} H_{{a, b}} = D_{{a, b}}$ ; applying this to $c$ and rearranging is the stated odd Jacobi identity.

Depends on

BV algebra with a degree-one generator

Used by

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Sources

Ezra Getzler, Batalin–Vilkovisky algebras and two-dimensional topological field theories, arXiv:hep-th/9212043v3 (standard reference, not scraped)
Paul Seidel, Lectures on Categorical Dynamics and Symplectic Topology, MIT Spring 2013 lecture notes (standard reference, not scraped)
Yvette Kosmann-Schwarzbach and Juan Monterde, Divergence operators and odd Poisson brackets, Annales de l’Institut Fourier 52 (2002) (standard reference, not scraped)