8.1 BV Algebras and the Master Equation
187
[n 1 + n 2 ] = S 1 S 2 , a permutation σ ∈ Σ n 1 +n 2 is the same as an isomorphism
S 1 S 2
∼ =
− → S 1 S 2 . Using (6.53) resp. its odd version we see that, for such a σ ,
M (σ )⊗T (σ )
a ◦ b ⊗ a • b
=
a ◦ b ⊗ a • b
M ( ˜
σ 1 )⊗M ( ˜
σ 2 )⊗T ( ˜
σ 1 )⊗T ( ˜
σ 2 )
,
where ˜
σ 1 : S 1 {a}
∼ =
− → σ (S 1 ) {a} is the isomorphism that restricts to σ on S 1 , and
˜
σ 2 : S 2 {b}
∼ =
− → σ (S 2 ) {b} is defined analogously. The above equality implies that
σ {g, h}
=
S 1 S 2 =[n 1 +n 2 ]
a ◦ b ⊗ a • b
τ
M ( ˜
σ 1 θ 1 ) ⊗ T ( ˜
σ 1 θ 1 ) ⊗ M ( ˜
σ 2 θ 2 ) ⊗ T ( ˜
σ 2 θ 2 )
(g ⊗ h).
Replacing the summation over S 1 , S 2 by the summation over σ (S 1 ), σ (S 2 ), and
the corresponding isomorphisms θ 1 , θ 2 by ˜
σ 1 θ 1 , ˜
σ 1 θ 2 we conclude that the last
expression equals {g, h} as desired.
Let us introduce the total graded vector space
MT :=
n≥0, g∈A
MT(n; g).
(8.4)
The operations defined in (8.1)–(8.3) act on sequences in MT, defining degree +1
operations d, Δ : MT → MT and {−, −} : MT ⊗ MT → MT.
Theorem 8.1 The object MT =
MT, d, Δ, {−, −}
is a desuspended bidifferential graded Lie algebra, i.e. the operations d : MT → MT, Δ : MT → MT
and {−, −} : MT ⊗ MT → MT have degree +1, {−, −} is graded symmetric and
the following axioms are fulfilled: 2
d
2
= Δd + dΔ = Δ
2
= 0,
(8.5)
d{f, g} + {df, g} + (−1)
|f |
{f, dg} = 0,
(8.6)
Δ{f, g} +
Δ(f ), g
+ (−1)
|f |
f, Δ(g)
= 0, and
(8.7)
(−1)
|f ||h|
{f, g}, h
+ (−1)
|h||g|
{h, f }, g
+ (−1)
|g||f |
{g, h}, f
= 0
(8.8)
for arbitrary homogeneous f, g, h ∈ MT.
2 Equivalently, the operations d, Δ and {−, −} induce on the suspension ↑ MT a dg-Lie algebra
structure.