8.2 Loop Homotopy Algebras
205
which, combined with the above calculations, shows that the map corresponding to
the left-hand side of (8.37) is given by
1≤i≤n−1
(−1)
|v 1 |+···+|v i−1 | δ
g
n−1 (v 1 , . . . , dv i , . . . , v n−1 ) + dδ
g
n−1 (v 1 , . . . , v n−1 ).
We conclude that the left-hand side of (8.37) is translated into d(δ
g
n−1 ), where d now
denotes the induced differential on the space of maps V ⊗n−1 → V .
The degree 2 map V ⊗(n−1) → V corresponding to the first term in the right-hand
side of (8.37) is, by (8.39), given as
(−1)
|s
j | f
g−1
n+2 (s
i , s
i , s
j , v 1 , . . . , v n−1 )s
j
=
(−1)
|s
j |+(|s
i |+|s
i |)|s
j | f
g−1
n+2 (s
j , s
i , s
i , v 1 , . . . , v n−1 )s
j
= −
δ
g−1
n+1 (s
i , s
i , v 1 , . . . , v n−1 ),
where we used that |s
j | + (|s
i | + |s
i |)|s
j | = |s
j | + |s
j | = 1.
The analysis of the second term in the right-hand side is subtler. Denote by
uSh (n 1 , n 2 ) the set of all (n 1 , n 2 )-unshuffles σ ∈ uSh(n 1 , n 2 ) such that σ (n) = n.
Notice that then
1
2
)(−1)
|s
i | f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )f
g 2
n 2 +1 (s
i , v σ (n 1 +1) , . . . , v σ (n) )
=
(σ )(−1)
|s
i | f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )
× f
g 2
n 2 +1 (s
i , v σ (n 1 +1) , . . . , v σ (n−1) , v n ),
where
denotes the sum in the first line of the display restricted to uSh
(n 1 , n 2 ).
Consequently, the degree +2 map V ⊗(n−1) → V corresponding to the second term
in the right-hand side of (8.37) is
−
)(−1)
|s
i |+|s
j |
f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )
× f
g 2
n 2 +1 (s
i , v σ (n 1 +1) , . . . , v σ (n−1) , s
j )s
j .
Noticing that
f
g 2
n 2 +1 (s
i , v σ (n 1 +1) , . . . , v σ (n−1) , s
j )s
j = (−1)
|s
j | δ
g 2
n 2 (s
i , v σ (n 1 +1) , . . . , v σ (n−1) ),
we rewrite it into
−
)(−1)
|s
i | f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )δ
g 2
n 2 (s
i , v σ (n 1 +1) , . . . , v σ (n−1) ).