88
B Homotopy Involutive Lie Bialgebras
is based. From a conceptual point of view, nothing new happens in the definition of
I BL ∞ -algebras when compared to the one of L ∞ - and A ∞ -algebras. The difference
is essentially that the underlying objects are more complicated. An I BL ∞ -algebra
is defined by an element L ∈ Coder(SA, x) of degree 1 that squares to zero:
L
2
= 0
and
|L| = 1 .
For completeness, we will now describe I BL-algebras as a special case of
I BL ∞ -algebras. Consider an element L ∈ Coder(SA, x) that consists of a strict
coderivation of order one and a strict coderivation of order two only:
L = L
1,0
+ xL
2,0 .
Furthermore, we restrict to the case where the only non-vanishing components of
l 1,0 := π 1 ◦ L 1,0 : SA → A and l 2,0 := π 2 ◦ L 2,0 : SA → A ∧2 are
d := l
1,0
◦i 1 : A → A , [−, −] := l
1,0
◦i 2 : A
∧2
→ A , δ := l
2,0
◦i 1 : A → A
∧2 .
To recover the definition of an involutive Lie bialgebra, we have to shift the degree
by one (see Appendix A), i.e., we define the operations on the shifted space ↑ A by
d :=↑ ◦d◦ ↓ ,
[−, −] := ↑ ◦[−, −] ◦ (↓)
∧2 , δ :=↑
∧2
◦ δ◦ ↓ .
The requirement L 2 = 0 is then equivalent to the following seven conditions
d
2
= 0,
(B.4)
d
[−, −] +
[−, −] (
d ∧ 1 + 1 ∧ d) = 0,
(B.5)
(
d ∧ 1 + 1 ∧
d) δ + δ
d = 0,
(B.6)
σ
[−, −] (
[−, −] ∧ 1) σ = 0,
(B.7)
σ σ ( δ ∧ 1 + 1 ∧ δ) δ = 0,
(B.8)
σ (
[−, −] ∧ 1) σ ( δ ∧ 1 + 1 ∧ δ) + δ
[−, −] = 0, and
(B.9)
[−, −] δ = 0.
(B.10)
In the above display, (B.4) means that d is a differential, (B.5) that
d is a derivation
for
[−, −], (B.6) that
d is a derivation for δ, (B.7) is the Jacobi identity for
[−, −], (B.8) is the co-Jacobi identity for δ, (B.9) is the compatibility between
δ and
[−, −], and (B.10) is the involutivity. We recognize the axioms defining a
differential involutive Lie bialgebra.