semifree dga in nLab

Contents

Contents

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 00 is of the form C ∞(X)C^\infty(X) for some space XX, which then is the space of objects of the Lie infinity-algebroid. But if it is a more general algebra in degree 00 one can think of a suitably generalized L ∞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 AA-coring (C,Δ,A)(C,\Delta, A) with a grouplike element gg associate its Amitsur complex with underlying graded module T A(Ω 1A)=⊕ n=0 ∞(Ω 1A) ⊗ AnT_A(\Omega^1 A)=\oplus_{n=0}^\infty (\Omega^1 A)^{\otimes_A n} where Ω 1=kerϵ\Omega^1=ker\,\epsilon and differential linearly extending the formulas da=ga−agd a = g a - a g for a∈Aa\in A and

dc=g⊗c+(−1) nc⊗g+∑ i=1 n(−1) ic 1⊗…⊗c i−1⊗Δ(c i)⊗c i+1⊗…⊗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⊗ A…⊗ Ac n∈(kerϵ) ⊗ Anc=c_1\otimes_A\ldots\otimes_A c_n\in (ker\,\epsilon)^{\otimes_A n};

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

(ag+ω)a′≔aa′g+ada′+ωa′. (a g +\omega)a' \coloneqq a a' g + a d a'+\omega a'.

In other words, we want the commutator [g,a′]=dω′[g,a']=d\omega'. We obtain an AA-bimodule. The coproduct on Ag⊕Ω 1AAg\oplus\Omega^1 A is Δ(ag)=ag⊗g\Delta(a g)=a g\otimes g and Δ(ω)=g⊗ω+ω⊗g−dω\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 11-11 correspondence with the comodules over the corresponding coring with a group-like element.

• 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]