The n-Category Café

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” $$Spaces \stackrel{\leftarrow}{\rightarrow} DGCAs$$ which relates spaces to DGCAs by sending a space $X$ to the DGCA $\Omega^\bullet(X)$ of forms on $X$ and sends a DGCA $A$ to the classifying space of flat $A$-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 $L_\infty$-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 $g$ 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 $g^*$ 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 $g$.

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.