206
8 Structures Relevant to Physics
Since f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i ) ∈ k, we use the multilinearity of δ
g 2
n 2 and further
rewrite the above sum into
−
)(−1)
|s
i | δ
g 2
n 2
f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )s
i , v σ (n 1 +1) , . . . , v σ (n−1)
.
Noticing finally that
f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )s
i = (−1)
|s
i | δ
g 1
n 1 (v σ (1) , . . . , v σ (n 1 ) ),
we see that the above sum equals
−
)δ
g 2
n 2
δ
g 1
n 1 (v σ (1) , . . . , v σ (n 1 ) ), v σ (n 1 +1) , . . . , v σ (n−1)
which in turn equals
−
)δ
g 2
n 2
δ
g 1
n 1 (v τ (1) , . . . , v τ (n 1 ) ), v τ (n 1 +1) , . . . , v τ (n−1)
,
where τ runs over all (n 1 , n 2 − 1)-unshuffles.
Summing the above calculations, we obtain the formula
0 = d(δ
g
n−1 )(v 1 , . . . , v n−1 ) +
δ
g−1
n+1 (s
i , s
i , v 1 , . . . , v n−1 )
(8.41)
+
)δ
g 2
n 2
δ
g 1
n 1 (v τ (1) , . . . , v τ (n 1 ) ), v τ (n 1 +1) , . . . , v τ (n−1)
.
that has to be satisfied for all homogeneous v 1 , . . . , v n−1 ∈ V . Recall that the second
summation in the right-hand side runs over all n 1 + n 2 = n, g 1 + g 2 = g, and
unshuffles τ ∈ uSh(n 1 , n 2 − 1). In a concise, elements-free form (8.41) reads
0 = d(δ
g
n−1 ) +
δ
g−1
n+1 (s
i , s
i , 1
⊗(n−1) ) +
δ
g 2
n 2 (δ
g 1
n 1 , 1
⊗(n 1 −1) )τ
−1 .
By the stability assumption, f 0
2 : V ⊗2 → V and therefore δ 0
1 : V → V is
identically zero. A useful trick is to define δ
0
1 := d to be the differential of the
underlying vector space. The first term in the right-hand side of (8.41) can then be
absorbed into the third one, leading to
0 =
)δ
g 2
n 2
δ
g 1
n 1 (v τ (1) , . . . , v τ (n 1 ) ), v τ (n 1 +1) , . . . , v τ (n−1)
(8.42)
+
δ
g−1
n+1 (s
i , s
i , v 1 , . . . , v n−1 ).
Changing the symbol n to n+1, n 1 to l, n 2 to k, g 1 to g 2 , g 2 to g 1 and τ to σ , and
replacing
s
i ⊗ s
i to
1
2
y i ⊗ y i we recognize the incarnation of loop homotopy
algebras described in [9, Sublemma 2]. Particular cases of this structure with δ
g
n = 0
8 Structures Relevant to Physics
Since f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i ) ∈ k, we use the multilinearity of δ
g 2
n 2 and further
rewrite the above sum into
−
)(−1)
|s
i | δ
g 2
n 2
f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )s
i , v σ (n 1 +1) , . . . , v σ (n−1)
.
Noticing finally that
f
g 1
n 1 +1 (v σ (1) , . . . , v σ (n 1 ) , s
i )s
i = (−1)
|s
i | δ
g 1
n 1 (v σ (1) , . . . , v σ (n 1 ) ),
we see that the above sum equals
−
)δ
g 2
n 2
δ
g 1
n 1 (v σ (1) , . . . , v σ (n 1 ) ), v σ (n 1 +1) , . . . , v σ (n−1)
which in turn equals
−
)δ
g 2
n 2
δ
g 1
n 1 (v τ (1) , . . . , v τ (n 1 ) ), v τ (n 1 +1) , . . . , v τ (n−1)
,
where τ runs over all (n 1 , n 2 − 1)-unshuffles.
Summing the above calculations, we obtain the formula
0 = d(δ
g
n−1 )(v 1 , . . . , v n−1 ) +
δ
g−1
n+1 (s
i , s
i , v 1 , . . . , v n−1 )
(8.41)
+
)δ
g 2
n 2
δ
g 1
n 1 (v τ (1) , . . . , v τ (n 1 ) ), v τ (n 1 +1) , . . . , v τ (n−1)
.
that has to be satisfied for all homogeneous v 1 , . . . , v n−1 ∈ V . Recall that the second
summation in the right-hand side runs over all n 1 + n 2 = n, g 1 + g 2 = g, and
unshuffles τ ∈ uSh(n 1 , n 2 − 1). In a concise, elements-free form (8.41) reads
0 = d(δ
g
n−1 ) +
δ
g−1
n+1 (s
i , s
i , 1
⊗(n−1) ) +
δ
g 2
n 2 (δ
g 1
n 1 , 1
⊗(n 1 −1) )τ
−1 .
By the stability assumption, f 0
2 : V ⊗2 → V and therefore δ 0
1 : V → V is
identically zero. A useful trick is to define δ
0
1 := d to be the differential of the
underlying vector space. The first term in the right-hand side of (8.41) can then be
absorbed into the third one, leading to
0 =
)δ
g 2
n 2
δ
g 1
n 1 (v τ (1) , . . . , v τ (n 1 ) ), v τ (n 1 +1) , . . . , v τ (n−1)
(8.42)
+
δ
g−1
n+1 (s
i , s
i , v 1 , . . . , v n−1 ).
Changing the symbol n to n+1, n 1 to l, n 2 to k, g 1 to g 2 , g 2 to g 1 and τ to σ , and
replacing
s
i ⊗ s
i to
1
2
y i ⊗ y i we recognize the incarnation of loop homotopy
algebras described in [9, Sublemma 2]. Particular cases of this structure with δ
g
n = 0