REPRESENTATIONS OF DERIVED A-INFINITY ALGEBRAS 3
2. Review of derived A∞-algebras
In this section we establish our notation and conventions. We review Sagave’s
definition of derived A∞-algebras from [Sag10] and we explain the operadic ap-
proach of [LRW13].
2.1. Derived A∞-algebras. Let k denote a commutative ring unless other-
wise stated. We start by considering (Z, Z)-bigraded k-modules
A =
i∈Z,j∈Z
Ai
j
.
We will use the following grading conventions. An element in Ai
j
is said to be
of bidegree (i, j). We call i the horizontal degree and j the vertical degree. We
have two suspensions:
(sA)i
j
=
Ai+1 j
and (SA)i
j
=
Ai+1.j
A morphism of bidegree (u, v) maps Ai
j
to Ai+u,
j+v
hence is a map of bidegree (0, 0)
s−vS−uA
→ A.
We remark that this is a different convention to that adopted in [LRW13].
The difference is a matter of changing the first grading from homological to coho-
mological conventions.
Note also that our objects are graded over (Z, Z). The reason for the change
will be explained below.
The following definition of (non-unital) derived A∞-algebra is that of [Sag10],
except that we generalize to allow a (Z, Z)-bigrading, rather than an (N,Z)-bigrading.
(Sagave avoids (Z, Z)-bigrading because of potential problems taking total com-
plexes, but this is not an issue for the purposes of the present paper.)
Definition 2.1. A derived A∞-algebra is a (Z,Z)-bigraded k-module A equipped
with k-linear maps
mij :
A⊗j
−→ A
of bidegree (−i, 2 − i − j) for each i ≥ 0, j ≥ 1, satisfying the equations
(2.1)
u=i+p,v=j+q−1
j=1+r+t
(−1)rq+t+pjmij(1⊗r
⊗ mpq ⊗
1⊗t)
= 0
for all u ≥ 0 and v ≥ 1.
Note that the map mij maps from
(A⊗j)α β
to
(A⊗j)α+2−i−j, β
−i
just as in [LRW13].
Thus the different convention for bidegrees has no effect on signs.
Examples of derived A∞-algebras include classical A∞-algebras, which are de-
rived A∞-algebras concentrated in horizontal degree 0. Other examples are bicom-
plexes, bidgas and twisted chain complexes (see below).
We remark that we follow the sign conventions of Sagave [Sag10]. For a derived
A∞-algebra concentrated in horizontal degree 0, one obtains one of the standard
sign conventions for A∞-algebras. The appendix contains a discussion of alternative
sign conventions, with a precise description of the relationship between them.
