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A A ∞ - and L ∞ -Algebras
In addition we have the obvious canonical projection maps π n : T A → A ⊗n and
the inclusion maps i n : A ⊗n → T A. A coderivation D ∈ Coder(T A) is a linear
map having the property
(D ⊗ 1 + 1 ⊗ D) ◦ Δ = Δ ◦ D .
( A . 1 )
The defining property (A.1) implies that a coderivation D ∈ Coder(T A) is uniquely
determined by a map d ∈ Hom(T A, A), i.e., Coder(T A) ∼ = Hom(T A, A).
Explicitly the correspondence reads
D ◦ i n =
i+j +k=n
1
⊗i
⊗ d j ⊗ 1
⊗k ,
where d n := d ◦ i n , 1 denotes the identity map on A and d = π 1 ◦ D. The
space of coderivations Coder(T A) turns out to be a Lie algebra with the Lie bracket
defined by
[D 1 , D 2 ] := D 1 ◦ D 2 − (−1)
D 1 D 2 D 2 ◦ D 1 .
An A ∞ -algebra is determined by a coderivation M ∈ Coder(T A) of degree 1
(degree −1 is considered if m 1 is supposed to be a boundary operator rather than a
coboundary operator) that squares to zero,
M
2
=
1
2
[M, M] = 0
and
|M| = 1 .
The corresponding homomorphism is defined by m := π 1 ◦ M.
In the case where only m 1 and m 2 are non-vanishing, we recover the definition
of a differential graded associative algebra up to a shift: Take A[1] =↑ A to be the
graded vector space defined by (↑ A) i = A i−1 . One has the map ↑: A →↑ A whose
only effect is increasing the degree by 1. Likewise, its inverse map ↓:↑ A → A
decreases the degree by one. The maps corresponding to the shifted space ↑ A are
defined by
˜
m n :=↑ ◦m n ◦ (↓)
⊗n
: (↑ A)
⊗n
→↑ A .
The operations ˜
m 1 and ˜
m 2 then determine a differential graded associative algebra,
if m n = 0 for n ≥ 3.
Consider now two A ∞ -algebras (A , M ) and (A , M ). An A ∞ -morphism F ∈
Morph(A , A ) from (A , M ) to (A , M ) is a degree 0 linear map F : T A →
T A satisfying
Δ ◦ F = (F ⊗ F ) ◦ Δ ,
F ◦ M
= M
◦ F.
(A.2)