nLab semifree dga

Definition

A differential graded algebra is semifree (or 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 of some (super)graded vector space.

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 .

Roiter’s theorem

Roiter’s theorem

• A. V. Roiter, Matrix problems and representations of BOCS’s; in Lec. Notes. Math. 831, 288–324 (1980)

says: 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

$$(a g +\omega)a' := a a' g + a d a'+\omega a'.$$

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.

Relation to Lie $\infty$-algebroids

One can identify semifree differential graded algebras in non-negative degree with Chevalley–Eilenberg algebras of (degreewise finite dimensional) Lie infinity-algebroids

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.

Terminology

Sometimes semi-free DGAs are called quasi-free, but this is in collision 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).

Related concepts

• Sullivan algebra

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