8.2 Loop Homotopy Algebras
201
for g ∈ T (n 1 + 1; g 1 )
Σ n 1 +1 and h ∈ T (n 2 + 2; g 2 )
Σ n 2 +1 , with the summation
running over all (n 1 , n 2 )-unshuffles σ ∈ Σ n 1 +n 2 , that is, permutations σ such that
σ (1) < · · · < σ (n 1 ) and σ (n 1 + 1) < · · · < σ (n 1 + n 2 ).
The master equation (8.20) describing morphisms α : F (QC # ) → T in terms
of degree 0 elements S ∈ QCT therefore reads
d T S = • 12 (S) +
1
2
σ ∈uSh(n 1 ,n 2 )
T (σ )(S n 1 +1 • 1 S).
By (8.33), S is a sequence of elements S(n; g) ∈ T (n; g) Σ n , (n, g) ∈ S sk ,
where the set S sk was defined in (6.71). It will be convenient to put by definition
S(n; g) := 0 if (n, g) ∈ S sk . The above master equation then means that, for each
(n, g) ∈ S sk ,
d T S(n; g) = • 12 S(n+2; g−1) +
1
2
S(n 1 +1; g 1 ) n 1 +1 • 1 S(n 2 +1; g 2 )
(8.36)
with the summation taken over n 1 +n 2 = n, g 1 +g 2 = g and σ ∈ uSh(n 1 , n 2 ).
Let us finally apply our machinery in the situation when T is the odd endomorphism operad End V of Example 6.31 in the skeletal presentation given in
Example 6.32. In this particular case we use more traditional notation and denote
f
g
n := S(n; g) ∈ End V (n; g), for n ≥ 0, g ≥ 0. So f
g
n is a degree 0 function
V ⊗n → k which is zero if (n, g) ∈ S sk . If V is equipped with a differential d, we
denote by the same symbol the differential induced in the standard manner on the
space of functions V ⊗n → k.
Theorem 8.3 An algebra over the Feynman transform F (QC # ) on a dg-vector
space V = (V , d) equipped with a degree +1 d-closed symmetric element
s =
s
i ⊗ s
i ∈ V ⊗ V
is the same as a collection
f
g
n : V
⊗n
→ k | (n, g) ∈ S sk
of degree 0 totally symmetric linear maps satisfying, for each (n, g) ∈ S sk , the
equation
d(f
g
n ) =
f
g−1
n+2 (s
i ⊗ s
i ⊗ 1
⊗n )
(8.37)
+
1
2
f
g 1
n 1 +1 (1
⊗n 1 ⊗ s
i )f
g 2
n 2 +1 (s
i ⊗ 1
⊗n 2 )
σ
−1 ,
201
for g ∈ T (n 1 + 1; g 1 )
Σ n 1 +1 and h ∈ T (n 2 + 2; g 2 )
Σ n 2 +1 , with the summation
running over all (n 1 , n 2 )-unshuffles σ ∈ Σ n 1 +n 2 , that is, permutations σ such that
σ (1) < · · · < σ (n 1 ) and σ (n 1 + 1) < · · · < σ (n 1 + n 2 ).
The master equation (8.20) describing morphisms α : F (QC # ) → T in terms
of degree 0 elements S ∈ QCT therefore reads
d T S = • 12 (S) +
1
2
σ ∈uSh(n 1 ,n 2 )
T (σ )(S n 1 +1 • 1 S).
By (8.33), S is a sequence of elements S(n; g) ∈ T (n; g) Σ n , (n, g) ∈ S sk ,
where the set S sk was defined in (6.71). It will be convenient to put by definition
S(n; g) := 0 if (n, g) ∈ S sk . The above master equation then means that, for each
(n, g) ∈ S sk ,
d T S(n; g) = • 12 S(n+2; g−1) +
1
2
S(n 1 +1; g 1 ) n 1 +1 • 1 S(n 2 +1; g 2 )
(8.36)
with the summation taken over n 1 +n 2 = n, g 1 +g 2 = g and σ ∈ uSh(n 1 , n 2 ).
Let us finally apply our machinery in the situation when T is the odd endomorphism operad End V of Example 6.31 in the skeletal presentation given in
Example 6.32. In this particular case we use more traditional notation and denote
f
g
n := S(n; g) ∈ End V (n; g), for n ≥ 0, g ≥ 0. So f
g
n is a degree 0 function
V ⊗n → k which is zero if (n, g) ∈ S sk . If V is equipped with a differential d, we
denote by the same symbol the differential induced in the standard manner on the
space of functions V ⊗n → k.
Theorem 8.3 An algebra over the Feynman transform F (QC # ) on a dg-vector
space V = (V , d) equipped with a degree +1 d-closed symmetric element
s =
s
i ⊗ s
i ∈ V ⊗ V
is the same as a collection
f
g
n : V
⊗n
→ k | (n, g) ∈ S sk
of degree 0 totally symmetric linear maps satisfying, for each (n, g) ∈ S sk , the
equation
d(f
g
n ) =
f
g−1
n+2 (s
i ⊗ s
i ⊗ 1
⊗n )
(8.37)
+
1
2
f
g 1
n 1 +1 (1
⊗n 1 ⊗ s
i )f
g 2
n 2 +1 (s
i ⊗ 1
⊗n 2 )
σ
−1 ,