A A ∞ - and L ∞ -Algebras
81
for any permutation σ ∈ Σ n , where Σ n denotes the permutation group of n
elements. Here, the elements c i are assumed to be of a definite degree and is
the Koszul sign. The comultiplication Δ : SA → SA ⊗ SA is defined by
Δ(c 1 , · · · , c n ) =
n
i=0
σ
(c σ 1 ∧ · · · ∧ c σ i ) ⊗ (c σ i+1 ∧ · · · ∧ c σ n ) ,
where
σ indicates the sum over all permutations σ ∈ Σ n constrained to
σ 1 < · · · < σ i and σ i+1 < · · · < σ n .
A coderivation D ∈ Coder(SA) is a linear map satisfying
(D ⊗ 1 + 1 ⊗ D) ◦ Δ = Δ ◦ D .
( A . 3 )
Again, the isomorphism Coder(SA) ∼ = Hom(SA, A) holds. The correspondence
between a coderivation D ∈ Coder(SA) and its associated map d = π 1 ◦ D ∈
Hom(SA, A) is given by
D ◦ i n =
i+j =n
σ
(d i ∧ 1
∧j ) ◦ σ ,
where σ in the right-hand side denotes the map that sends c 1 ∧ · · · ∧ c n into
(−1) c σ 1 ∧ · · · ∧ c σ n (again d n = d ◦ i n and 1 is the identity map on A).
An L ∞ -algebra is determined by a coderivation L ∈ Coder(SA) of degree 1 that
squares to zero,
L
2
= 0
and
|L| = 1 .
An L ∞ -morphism F ∈ Morph(A , A ) from an L ∞ -algebra (A , L ) to another
L ∞ -algebra (A , L ) is a degree 0 linear map F : SA → SA such that
Δ ◦ F = (F ⊗ F ) ◦ Δ ,
F ◦ L
= L
◦ F.
(A.4)
Such an F is determined by a map f = π 1 ◦ F ∈ Hom(SA, A ) through
F =
∞
n=0
1
n!
f
∧n
◦ Δ n ,
( A . 5 )
where Δ n : SA → (SA) ⊗n denotes the (n−1)-fold comultiplication.