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August 17, 2008

Differential Graded Clifford Algebra

Posted by Urs Schreiber

Take graded Grassmann algebras and equip them with a differential of degree +1 squaring to 0. The result, quasi-free differential graded algebras, happens to be equivalent to \infty-Lie algebras whose bracket \infty-functor is strictly skew symmetric but satisfies the Jacobi identity only up to higher coherent equivalence.

What natural generalizations of this are available such that skew symmetry is relaxed? There is an obvious one: Grassmann algebra generalizes to Clifford algebra if skew-symmetry is relaxed, the failure of skew symmetry being measured by a bilinear symmetric form.

I had suggested that we should generalize everything we are doing with quasi-free differential graded-commutative algebras to differential graded Clifford algebras.

For instance the “differential homotopy hypothesis” SpacesDGCAs Spaces \stackrel{\leftarrow}{\rightarrow} DGCAs which relates spaces to DGCAs by sending a space XX to the DGCA Ω (X)\Omega^\bullet(X) of forms on XX and sends a DGCA AA to the classifying space of flat AA-valued differential forms would be generalized to a “Riemannian homotopy hypothesis” where a Riemannian space is sent to the graded Clifford algebra of sections of the deRham spinor bundle with differential given by the Dirac operator. On the other side we’d consider generalizations of the familiar Chevalley-Eilenberg and Weil DGCAs of L L_\infty-algebras to deformations where graded symmetric bilinear forms deform the graded-symmetric wedge products to a graded Clifford product.

Now, as Hisham Sati kindly points out to me, aspects of the latter have already been considered in the literature before. For gg an ordinary Lie algebra with non-degenarate bilinear invariant form ,\langle \cdot, \cdot \rangle , the article

A. Alekseev and E. Meinrenken
The non-commutative Weil algebra
arXiv:math/9903052

considers the deformation of the Chevalley-Eilenberg algebra from a differential graded commutative algebra to a differential graded Clifford algebra with the Clifford product controled by ,\langle \cdot , \cdot \rangle. Notice that the same kind of bilinear invariant forms control Dmitry Roytenberg’s weak Lie 2-algebras. And notice that due to the grading this amounts to regarding the Clifford pairing to be of degree -2 (sending two degree 1 generators to a scalar in degree 0).

On top of that, this article deforms also the corresponding Weil algebra (well, in fact the authors do not consider the CE-algebra separately, I just do this here to amplify the pattern). Now, in that the shifted copy of g *g^* appears in degree 2 and hence a degree -2 Clifford structure here is an skew bracket. The natural such bracket in the game is the original Lie bracket, so they take the shifted part of the Weil algebra to be the universal enveloping algebra of gg.

The resulting structure is not unrelated to what happens in BV-formalism with the introduction of the anti-fields: the ordinary differential on the Weil algebra still is a differential on its Clifford-deformation, but there now it is “Hamiltonian” in the sense that its action is given by a graded commutator with an element of the algebra – that element is a kind of Dirac operator. Its square is the quadratic Casimir operator.

Lots of nice things to be explored here. But I’ll leave it at that for the moment.

Posted at August 17, 2008 12:04 PM UTC

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Re: Differential Graded Clifford Algebra

For instance the “differential homotopy hypothesis”…which relates spaces to DGCAs

Hi Urs, can you remind us again precisely what the differential homotopy hypothesis (and the other variants) are saying? Is it the statement ‘This correspondence is an adjunction’ or is it something else?

Posted by: Bruce Bartlett on August 17, 2008 5:37 PM | Permalink | Reply to this

Re: Differential Graded Clifford Algebra

For instance the “differential homotopy hypothesis”…which relates spaces to DGCAs

Hi Urs, can you remind us again precisely what the differential homotopy hypothesis (and the other variants) are saying? Is it the statement ‘This correspondence is an adjunction’ or is it something else?

Right, I was being sloppy and didn’t introduce my terminology.

Given some notion of “Spaces” and some notion of \inftyGroupoids the corresponding homotopy hypothesis relates the procedure XΠ (X)X \mapsto \Pi_\infty(X) “send a space to its fundamental \infty-groupoid” with the procedure C|C|C \mapsto |C| “send an \infty-groupoid to its classifying space”. Depending on which models one uses, the “homotopy hypothesis” may be either of a) formulated and proven, b) formulated and conjectural, c) not even formulated yet, I gather.

There is a setup where both the spaces and the \infty-groupoids involved are smooth in some sense. One then observes that there is a “differential” or “infinitesimal” version of the fundamental \infty-groupoid Π (X)\Pi_\infty(X) of a smooth space XX: this is the tangent Lie algebroid TXT X, incarnated preferably in terms of its Chevalley-Eilenberg algebra Ω (X)\Omega^\bullet(X), the deRham DGCA of forms on XX.

Moreover, there is a notion of smooth classifying space of flat forms with values in a DGCA. If the DGCA is the Chevalley-Eilenberg algebra of an L L_\infty-algebroid gg this is the classifying space S(CE(g))S(CE(g)) of flat gg-valued forms.

So this setup

SpacesΩ ()S()DGCAsCEL Spaces \stackrel{\stackrel{S(-)}{\leftarrow}}{\stackrel{\Omega^\bullet(-)}{\rightarrow}} DGCAs \stackrel{CE}{\leftarrow} L_\infty

is the “differential” or “infinitesimal” or “Lie” version of the “smooth homotopy hypothesis”. In need of a term, I started thinking of it as the “differential homotopy hypothesis”.

And, yes, thanks to Todd Trimble we know that SS and Ω \Omega^\bullet form an adjunction.

Posted by: Urs Schreiber on August 17, 2008 8:25 PM | Permalink | Reply to this

Re: Differential Graded Clifford Algebra

Hi Urs,

What confuses me is: what is the actual hypothesis in the ‘differential homotopy hypothesis’? Right, very nice, we have these arrows going left and right… but what is the hypothesis saying?

The ‘homotopy hypothesis’, as you sketched above, is a claim that the arrows you mentioned are an equivalence of gismos

(1)SpacesGroupoids. Spaces \leftrightarrow \infty-Groupoids.

I’m wondering what is the corresponding claim in the differential homotopy hypothesis.

Posted by: Bruce Bartlett on August 17, 2008 9:46 PM | Permalink | Reply to this

Re: Differential Graded Clifford Algebra

what is the actual hypothesis in the ‘differential homotopy hypothesis’?

I also need a word for the setup itself. Spaces here, (infinitesimal) paths in spaces there. Is that the “homotopy setup”?

But I think the hypothesis always says “there is a suitable notion of equivalence” and then the task is to figure out what suitable means.

We have already that adjunction spaces-DGCAs. Now the obvious guess would be that something closer to an equivalence is obtained by restricting DGCAs to the image of the map L CEDGCAsL_\infty \stackrel{CE}{\to} DGCAs, i.e. to DGCAs which are quasi free over their degree 0-part as graded commutative algebras.

I think what one really would like to see in the end is a Quillen model equivalence. There is a well-known model structure on differential graded-commutative algebroids, i.e. on DG-categories, e.g. slide 15 here. This should induce a model structure on DGCAs and on L L_\infty-algebroids, I’d hope. But I don’t know. Does anyone?

Posted by: Urs Schreiber on August 17, 2008 10:05 PM | Permalink | Reply to this

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