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B Homotopy Involutive Lie Bialgebras
There are two ways to define higher order coderivations. One is based on
algebraic relations like that in Eq. (A.3). A coderivation of order two is, for example,
characterized by
Δ 3 ◦ D −
σ
σ ◦ (Δ ◦ D ⊗ 1) ◦ Δ +
σ
σ ◦ (D ⊗ 1
⊗2 ) ◦ Δ 3 = 0 ,
where
σ denotes the sum over inequivalently acting permutations in Σ 3 , the
permutation group of three elements, and σ : SA ⊗3 → SA ⊗3 is the map that
permutes the three factors. For completeness we state an algebraic definition of a
coderivation D ∈ Coder
n (SA) of order n,
n
i=0
σ
(−1)
i σ ◦ (Δ n+1−i ◦ D ⊗ 1
⊗i ) ◦ Δ i+1 = 0 .
(B.2)
As in the case of a coderivation of order one, this relation is saying—and this is
an alternative definition of higher order coderivations—that a coderivation D ∈
Coder
n (SA) of order n is uniquely determined by a map d ∈ Hom(SA, Σ n A),
where Σ n A = ⊕
n
i=0 A ∧i . Thus in contrast to a coderivation of order one, a
coderivation of order n is determined by a linear map on SA with n and less outputs
rather than just one output. The explicit relation between D ∈ Coder
n (SA) and
d ∈ Hom(SA, Σ n A) is
D ◦ i n =
i+j =n
σ
(d i ∧ 1
∧j ) ◦ σ ,
which is a naive generalization of Eq. (B.1).
A trivial observation is that a coderivation of order n − 1 is also a coderivation of
order n, by simply defining the component with n outputs to be zero, that is,
Coder
n−1 (SA) ⊂ Coder
n (SA) .
We call a coderivation D ∈ Coder
n (SA) of order n a strict coderivation of order n
if the corresponding map d is in Hom(SA, A ∧n ), that is, if the map d has exactly n
outputs. In that case we can identify d = π n ◦ D.
Next recall the graded commutator
[D 1 , D 2 ] = D 1 ◦ D 2 − (−1)
D 1 D 2 D 2 ◦ D 1 ,
where D 1 , D 2 are arbitrary higher order coderivations. Using the defining equations
(B.2), it can be shown that
[Coder
i (SA), Coder
j (SA)] ⊂ Coder
i+j −1 (SA) .
( B . 3 )