B Homotopy Involutive Lie Bialgebras
87
In the case i = j = 1 we recover that [−, −] defines a Lie algebra on Coder
1 (SA),
but we see that [−, −] does not define a Lie algebra at higher orders n > 1. Of
course, we can make the collection of all higher order coderivations a Lie algebra,
but in the next section we will see that there is still a finer structure.
B.2
IBL ∞ -Algebras
Now we have all tools to define I BL ∞ -algebras. We will furthermore see that one
recovers an involutive Lie bialgebra (I BL-algebra) as a special case of an I BL ∞ -
algebra. Consider the space
Coder(SA, x) :=
∞
n=1
x
n−1 Coder
n (SA) ,
where x ∈ k is some auxiliary parameter. An element D ∈ Coder(SA, x) can be
expanded as
D =
∞
n=1
x
n−1 D
(n) ,
where D (n) ∈ Coder
n (SA). In the following, we will indicate coderivations of order
n by the superscript (n) and strict coderivations of order n by the superscript n.
We can decompose every coderivation of order n into strict coderivations of order
smaller than or equal to n. Accordingly, we denote the strict coderivation of order
n−g corresponding to a coderivation D (n) of order n by D n−g,g , g ∈ {0, . . . , n−1}
(in the main text g was identified as the genus). Thus, we have
D
(n)
=
n−1
g=0
D
n−g,g ,
and D expressed in terms of strict coderivations reads
D =
∞
n=1
∞
g=0
x
n+g−1 D
n,g .
Due to Eq. (B.3), we have
[D 1 , D 2 ] ∈ Coder(SA, x) ,
that is, the commutator [−, −] turns Coder(SA, x) into a graded Lie algebra. The
space Coder(SA, x) is the Lie algebra on which our definition of I BL ∞ -algebras