210
8 Structures Relevant to Physics
A Lie bialgebra as above is involutive if, moreover
[a
i , a
i ] = 0,
(8.51)
for a ∈ V and
a
i ⊗ a
i := δa.
Returning to an IBL ∞ -algebra in the beginning of this example, one may define
degree 0 linear operations on the desuspension V =↓ U by
:=↓ ω
1
2,2,0 (↑⊗ ↑) : V ∧ V → V and δ := (↓⊗ ↓) ω
2
1,1,0 ↑: V → V ∧ V .
Due to the special form (8.48) of Δ, one has
Δ
2
= (Δ 1 + ¯
hΔ 2 )
2
= Δ
2
1 + (Δ 1 Δ 2 + Δ 2 Δ 1 ) ¯
h + Δ
2
2 ¯
h
2 .
Therefore Δ 2 = 0 is equivalent to the separate vanishing of Δ 2
1 , (Δ 1 Δ 2 + Δ 2 Δ 1 ) 2 ,
and Δ 2
2 . Since Δ 2
1 is a derivation, see [9, Proposition 1], its vanishing is equivalent
to the vanishing of the restriction Δ 2
1 | U : U → S 3 (U ) which is easily seen to
be equivalent to the dual Jacobi identity for δ. A similar reasoning shows that the
vanishing of (Δ 1 Δ 2 + Δ 2 Δ 1 ) 2 is equivalent to the compatibility (8.50) and the
involutivity (8.51). Finally, the vanishing of Δ 2
2 is equivalent to the Jacobi identity
for [−, −] = . With this example in mind, one might view IBL ∞ -algebras as
homotopy versions of involutive Lie bialgebras, which explains the terminology.
Example 8.3 In this and the following example we assume that U is a graded vector
space with finite-dimensional components. With this assumption, IBL ∞ -algebras
whose only nontrivial operations are
ω
s
1,1,0 : U → S
s (U ), s ≥ 1,
are the same as L ∞ -algebras on the dual V = U # of U . Indeed, the operator Δ
in (8.43) is in this case just a derivation Δ 1 : S(U ) → S(U ) such that Δ 2
1 = 0.
Our statement is then the second part of [8, Theorem 2.3] with the reversed grading.
Notice that d := (ω 1
1,1,0 ) # : V → V is a degree +1 differential.
Example 8.4 We leave as an exercise to prove that IBL ∞ -algebras whose only
nontrivial operations are
ω
1
1,1,0 : U → U,
ω
s
1,1,g : U → S
s (U )
s≥2,g≥0
and B := ω
0
2,2,0 : S
2 (U ) → k
(8.52)
are the same as loop homotopy Lie algebras on the dual V := U # in the form (8.42).
Indeed, defining the differential d on V to be the dual of ω 1
1,1,0 , the map s : k →
S 2 (V ) the dual of B and the operations δ
g
n the duals of ω
s
1,1,g , we see that the
8 Structures Relevant to Physics
A Lie bialgebra as above is involutive if, moreover
[a
i , a
i ] = 0,
(8.51)
for a ∈ V and
a
i ⊗ a
i := δa.
Returning to an IBL ∞ -algebra in the beginning of this example, one may define
degree 0 linear operations on the desuspension V =↓ U by
:=↓ ω
1
2,2,0 (↑⊗ ↑) : V ∧ V → V and δ := (↓⊗ ↓) ω
2
1,1,0 ↑: V → V ∧ V .
Due to the special form (8.48) of Δ, one has
Δ
2
= (Δ 1 + ¯
hΔ 2 )
2
= Δ
2
1 + (Δ 1 Δ 2 + Δ 2 Δ 1 ) ¯
h + Δ
2
2 ¯
h
2 .
Therefore Δ 2 = 0 is equivalent to the separate vanishing of Δ 2
1 , (Δ 1 Δ 2 + Δ 2 Δ 1 ) 2 ,
and Δ 2
2 . Since Δ 2
1 is a derivation, see [9, Proposition 1], its vanishing is equivalent
to the vanishing of the restriction Δ 2
1 | U : U → S 3 (U ) which is easily seen to
be equivalent to the dual Jacobi identity for δ. A similar reasoning shows that the
vanishing of (Δ 1 Δ 2 + Δ 2 Δ 1 ) 2 is equivalent to the compatibility (8.50) and the
involutivity (8.51). Finally, the vanishing of Δ 2
2 is equivalent to the Jacobi identity
for [−, −] = . With this example in mind, one might view IBL ∞ -algebras as
homotopy versions of involutive Lie bialgebras, which explains the terminology.
Example 8.3 In this and the following example we assume that U is a graded vector
space with finite-dimensional components. With this assumption, IBL ∞ -algebras
whose only nontrivial operations are
ω
s
1,1,0 : U → S
s (U ), s ≥ 1,
are the same as L ∞ -algebras on the dual V = U # of U . Indeed, the operator Δ
in (8.43) is in this case just a derivation Δ 1 : S(U ) → S(U ) such that Δ 2
1 = 0.
Our statement is then the second part of [8, Theorem 2.3] with the reversed grading.
Notice that d := (ω 1
1,1,0 ) # : V → V is a degree +1 differential.
Example 8.4 We leave as an exercise to prove that IBL ∞ -algebras whose only
nontrivial operations are
ω
1
1,1,0 : U → U,
ω
s
1,1,g : U → S
s (U )
s≥2,g≥0
and B := ω
0
2,2,0 : S
2 (U ) → k
(8.52)
are the same as loop homotopy Lie algebras on the dual V := U # in the form (8.42).
Indeed, defining the differential d on V to be the dual of ω 1
1,1,0 , the map s : k →
S 2 (V ) the dual of B and the operations δ
g
n the duals of ω
s
1,1,g , we see that the