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semifree dga

Definition

A differential graded algebra is semifree (or semi-free) if, after forgetting the differential, it is isomorphic as a graded algebra to a tensor (polynomial) algebra of some (super)vector space.

A differential graded-commutative algebra is semifree (or semi-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 $T_A({\Omega }^{1}A)={\oplus }_{n=0}^{\mathrm{\infty }}({\Omega }^{1}A{)}^{{\otimes }_{A}n}$ where ${\Omega }^1=\ker ϵ$ and differential linearly extending the formulas $da=ga-ag$ for $a\in A$ and

$$dc=g\otimes c+(-1{)}^{n}c\otimes g+\sum _{i=1}^n(-1{)}^i{c}_1\otimes \dots \otimes c_{i-1}\otimes \Delta (c_i)\otimes c_{i+1}\otimes \dots \otimes c_n$$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 $c=c_1{\otimes }_A\dots {\otimes }_A{c}_n\in (\ker ϵ{)}^{{\otimes }_{A}n}$;

conversely, to a semi-free dga ${\Omega }^{•}A$ one associates the $A$-coring $Ag\oplus {\Omega }^{1}A$ 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

$$(ag+\omega )a\prime :=aa\prime g+ada\prime +\omega a\prime .$$(a g +\omega)a' := a a' g + a d a'+\omega a'.

In other words, we want the commutator $[g,a\prime ]=d\omega \prime$. We obtain an $A$-bimodule. The coproduct on $\mathrm{Ag}\oplus {\Omega }^{1}A$ is $\Delta (ag)=ag\otimes g$ 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 $\mathrm{\infty }$-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 $C^{\mathrm{\infty }}(X)$ 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 $L_{\mathrm{\infty }}$-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).