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 Aj. i We will use the following grading conventions. An element in Aj i is said to be of bidegree (i, j). We call i the horizontal degree and j the vertical degree. We have two suspensions: (sA)j i = Aj+1 i and (SA)j i = Aj i+1 . A morphism of bidegree (u, v) maps Aj i to Aj+v, i+u 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)β+2−i−j, α−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.
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