nLab semifree dga

Definition

A differential graded algebra over some ground field (or ground ring) is called semi-free if the underlying graded algebra is free: if after forgetting the differential, it is isomorphic as a graded algebra to a (polynomial) tensor algebra over the ground field (ground ring) of some (super)graded vector space (or graded module, or bimodule if the ground ring is not commutative – in this generality see Roiter 1980 p 296).

A differential graded-commutative algebra is semifree (or semi-free) if the underlying graded-commutative algebra is free: if after forgetting the differential, it is isomorphic as a graded-commutative algebra to a Grassmann algebra of some graded vector space .

Sometimes semi-free DGAs are called quasi-free, but this clashes with the terminology about formal smoothness of noncommutative algebras, i.e. quasi-free algebras in the sense of Cuntz and Quillen (and with extensions to homological smootheness of dg-algebras by Kontsevich).

Properties

Relation to Lie $\infty$-algebroids

One may identify semifree differential graded algebras with Chevalley-Eilenberg algebras of (degreewise finite dimensional) L-infinity algebroids (Sc08, SSS12), generalizing a corresponding statement for L-infinity algebras (see there).

At least when the algebra in degree $0$ is of the form $C^\infty(X)$ for some space $X$, which then is the space of objects of the Lie infinity-algebroid. But if it is a more general algebra in degree $0$ one can think of a suitably generalized $L_\infty$-algebroid, for instance with a noncommutative space of objects. This generalizes the step from Lie algebroids to Lie-Rinehart pairs.

Roiter’s theorem

The main theorem of Roiter 1980 says that semi-free differential graded algebras are in bijective correspondence with corings with a grouplike element:

to an $A$-coring $(C,\Delta, A)$ with a grouplike element $g$ associate its Amitsur complex with underlying graded module $T_A(\Omega^1 A)=\oplus_{n=0}^\infty (\Omega^1 A)^{\otimes_A n}$ where $\Omega^1=ker\,\epsilon$ and differential linearly extending the formulas $d a = g a - a g$ for $a\in A$ and

$$d c = g\otimes c + (-1)^n c\otimes g +\sum_{i=1}^n (-1)^i c_1\otimes\ldots\otimes c_{i-1}\otimes\Delta(c_i)\otimes c_{i+1}\otimes\ldots\otimes c_n$$

for $c=c_1\otimes_A\ldots\otimes_A c_n\in (ker\,\epsilon)^{\otimes_A n}$;

conversely, to a semi-free dga $\Omega^\bullet A$ one associates the $A$-coring $A g\oplus\Omega^1 A$ where $g$ isa new group-like indeterminate; this is by definition a direct sum of left $A$-modules with a right $A$-module structure given by

$$(a g +\omega)a' \coloneqq a a' g + a d a'+\omega a'.$$

In other words, we want the commutator $[g,a']=d\omega'$. We obtain an $A$-bimodule. The coproduct on $Ag\oplus\Omega^1 A$ is $\Delta(a g)=a g\otimes g$ and $\Delta(\omega)= g\otimes\omega+\omega\otimes g- d\omega$. The two operations are mutual inverses (see lectures by Brzezinski or the arxiv version math/0608170).

Moreover flat connections for a semi-free dga are in $1$-$1$ correspondence with the comodules over the corresponding coring with a group-like element.

Related concepts

• Sullivan algebra

• L-infinity algebra, super L-infinity algebra, “FDA”

References

Roiter’s theorem:

• A. V. Roiter: Matrix problems and representations of BOCS’s, in Lec. Notes. Math. 831 (1980) 288-324 [doi:10.1007/BFb0089782]

In relation to L-infinity algebroids (and specifically of L-infinity algebras, see there):

• Urs Schreiber: On $\infty$-Lie (2008) [pdf, Schreiber-InfinityLie.pdf]

• Hisham Sati, Urs Schreiber, Jim Stasheff Section A.1 of: Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics 315 1 (2012) 169-213 [arXiv:0910.4001, doi:10.1007/s00220-012-1510-3]