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.