Categorified Clifford Algebra and weak Lie n-Algebras

October 9, 2007

Categorified Clifford Algebra and weak Lie n-Algebras

Posted by Urs Schreiber

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What is a categorified Grassmann algebra?

What is a categorified Clifford algebra?

What differential algebraic structure are fully weak Lie $n$ -algebras equivalent to?

Is there a relation between these questions?

The principle of least resistance under categorification says

We understand the true nature of a concept the deeper, the more straightforwardly the definition we use to conceive it lends itself to categorification.

Hence we have a complete understanding of the true meaning of the concept of a Lie group. And therefore still a rather good understanding of the concept of Lie algebra.

But do we already, in this sense, understand the true nature of the concepts “Grassmann algebra” and “Clifford algebra”?

Of course I could try to describe a categorified Grassmann algebra as something like an abelian monoidal category equipped with a categorified version of graded-commutativity.

But it turns out that there is something even less resistive:

A Grassmann algebra $\wedge^\bullet V$ over a vector space $V$ is related by Koszul duality to the abelian Lie algebra on $V$ .

(See for instance the beginning of Lie $n$ -algebra cohomology for more on how this works.)

But we said Lie algebras are nicely categorified. So we should maybe say

An $n$ -Grassmann algebra $\wedge^\bullet V$ over a vector space $V$ is defined to be the Koszul dual to an abelian semistrict Lie $n$ -algebra.

That would imply that an $n$ -Grassmann algebra is the graded-commutative algebra

$\wedge^\bullet V$

freely generated over a graded vector space concentrated in degrees $1 \leq d \leq n$ .

One generalization of this fact is well known: as we pass from abelian to general Lie $n$ -algebras – whose bracket is strictly skew-symmetric but whose Jacobi identity holds only up to coherent equivalence – the Koszul-dual algebraic side generalizes from free graded-commutative algebras to differential graded commutative algebras.

In fact, people use precisely this kind of identification to set up their definitions: since on the side of differential graded algebras the generalization to many-objects is obvious, one defines a Lie $n$ -algebroid to be (dual to) a suitable dg-manifold.

This means we are left with two open questions:

- we still need to figure out what happens as we replace Grassmann algebras by Clifford algebras here

- we are still assuming that the skew-symmetry of the bracket functor $[\cdot,\cdot] : S \times S \to S$ holds strictly.

In Detecting Higher Order Necklaces I conjectured that these two items are indeed dual to each other.

If true, this would mean that

An $n$ -Clifford algebra over a vector space $V$ is defined to be the Koszul dual to an abelian fully weak Lie $n$ -algebra.

and presumeably that

Fully weak Lie $n$ -algebras are Koszul dual to differential graded Clifford algebras.

Today mankind made one further step towards checking this conjecture: Dmitry Roytenberg has now issued his thoughts on fully weak Lie 2-algebras:

Dmitry Roytenberg
On weak Lie 2-algebras
(pdf)

Exercise: Give the codifferential coalgebra description of Dmitry Roytenberg’s weak 2-term $L_\infty$ -algebras (p. 9). Then dualize to find the corresponding differential algebra. Check if it can be sensibly addressed as a differential graded Clifford algebra.

(Notice that we expect to see “graded” Clifford algebra: the anticommutator of two degree 1 elements is degree 2.)

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Reminder of some of the underlying facts

John Baez and Alissa Crans essentially defined Lie $n$ -algebras to be $(n-1)$ -categories $S$ internal to vector spaces, equipped with a product functor

$[\cdot, \cdot] : S \times S \to S$

which is strictly skew symmetric and satisfies the Jacobi identiy up to coherent equivalence.

Abstract operad nonsense shows that such a “semistrict” Lie $n$ -algebra is the same a an $n$ -ter, $L_\infty$ -algebra, i.e. an $L_\infty$ on a vector space $V$ which is concentrated in degree $1 \leq p \leq n$ .

For more details see for instance the beginning of Lie $n$ -algebra cohomology and the references given there.

Straightforward computation, in turn, shows that $n$ -term $L_\infty$ algebras are the same as free graded commutative coalgebras $S^c V$ equipped with a degree -1 codifferential

$D : S^c V \to S^c V$

of degree -1 such that

$D^2 = 0 \,.$

For $V$ finite, dualizing this statement leads to the statement that Lie $n$ -algebras are equivalently encoded in graded Grassmann algebras

$\wedge^\bullet V^*$

equipped with a differential

$d : \wedge^\bullet V^* \to \wedge^\bullet V^*$

of degree +1 and such that

$d^2 = 0 \,.$

Posted at October 9, 2007 4:01 PM UTC

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8 Comments & 4 Trackbacks

Re: Categorified Clifford Algebra and weak Lie n-Algebras

Any idea what the categorification of the Clifford algebra clock looks like? A categorified clock of categorified Clifford algebras?

Hmm, how to categorify a clock?

Posted by: David Corfield on October 11, 2007 9:18 AM | Permalink | Reply to this

Re: Categorified Clifford Algebra and weak Lie n-Algebras

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Any idea what the categorification of the Clifford algebra clock looks like?

No, unfortunately not. Wish I had more time at the moment thinking this through.

There is another thing which would be interesting to think about further:

what would Clifford algebras integrate to, if we regard them as Lie 2-algebra with weak skew symmetry?

Well, the ordinary skew symmetry of the Lie bracket derives, when you obtain it from a Lie group, from the properties of the group commutator. Its antisymmetry translates into the fact that inverses in a group are strict. So, I think, a Lie $n$ -algebra with weak skew symmetry integrates to a Lie $n$ -groupoid with weak inverses.

So consider the ordinary additive Lie group of real numbers. We can think of any of its group elements $t$ as a homotopy class of paths on the real line (a morphism in $\Pi_1(\mathbb{R})$ ): represented by any path starting at the origin and ending a distance $t$ away from it.

But then let’s pass from homotopy classes of paths on the line to parameterized paths on the line (Moore paths). Composition is still strictly associative, but now there are no inverses. But we can create weak inverses by thrwoing in 2-morphisms that connect the constant path with any path that has coinciding source and target.

So that means we pass to $\Pi_2(\mathbb{R})$ ! The Lie 2-group obtained this way has strictly associative composition of 1-morphisms, but inverses of 1-morphisms exist only up to (unique) 2-ismorphism.

This should differentiate to a Lie 2-algebra with strict Jacobi identity satisfied by skew symmetry being weakened. In fact, I expect that the algebra Koszul dual to this Lie 2-algebra is essentially $\mathrm{Cl}_1$ .

Posted by: Urs Schreiber on October 11, 2007 11:11 AM | Permalink | Reply to this

Re: Categorified Clifford Algebra and weak Lie n-Algebras

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David wrote:

Any idea what the categorification of the Clifford algebra clock looks like?

Andre Henriques is working on a quite different concept of ‘categorified Clifford algebra’ which is supposed to fill in this analogy:

K-theory : Clifford algebras :: elliptic cohomology : ???

Real K-theory has period 8, and that ‘Bott periodicity’ comes from the 8 hours of the Clifford algebra clock. Elliptic cohomology has period 24 — or something like that — and this periodicity should come from the 24 hours of the ??? clock.

Of course it’s quite nice for a clock to have 24 hours.

I say “or something like that”, because modular forms display a period-12 phenomenon, while topological modular forms (the sophisticated way of thinking about elliptic cohomology) actually display a period- $576$ phenomenon, where $576 = 24 \times 24$ — but I remember Henriques talking about period 24.

Posted by: John Baez on October 13, 2007 8:38 AM | Permalink | Reply to this

Re: Categorified Clifford Algebra and weak Lie n-Algebras

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quite different concept

They might be related.

Next step for me here would be to formulate Dirac operators from this point of view.

These are “quantized connections”, or rather “quantized covariant derivatives”, where the “quantization” means replacement of Grassmann by Clifford algebra.

But that’s precisely what the proposal here is about.

I am not entirely sure yet, but am thinking that we should be able to regard Dirac operators as covariant derivatives for Lie-algebras with weak skew-symmetry.

If this works, it would be weird if covariant derivatives for weak Lie 2-algebras weren’t the right notion of 2-Dirac operator.

So, let’s see.

For $X = \mathbb{R}^n$ , say, the pair groupoid $X \times X$ gives rise to the tangent Lie algebroid $T X$ and the dually to differential forms $\Omega^\bullet(X)$ .

Now weaken the pair groupoid: keep associativity but make inverses weak. So now 1-morphisms are sequences of points in $X$ $(x_1 , x_2 , \cdots , x_n)$ and composition is simply concatenation. On top of that, we have $mathbb{R}$ -worth of 2-morphisms filling triangles of the form $\array{ && y \\ &\nearrow &\Downarrow^\eta& \searrow \\ x &&& x }$

The weak invertator $\eta$ we take to be $\eta(x,y) := |x-y|^2$ for a fixed chosen metric.

I am speculating that if we now take the Lie 2-algebroid and then dualize we get not the exterior but the Clifford algebra corresponding to the chosen metric.

For an ordinary $g$ -connection $\Omega^\bullet(X) \stackrel{(A,F_A)}{\leftarrow} \mathrm{inn}(g)^*$ a function and its covariant derivative is given by a choice of transformation

$\array{ & \swarrow \nwarrow^0 \\ \Omega^\bullet(X) &\Downarrow^{(e,\nabla e)}& \mathrm{inn}(g)^* \\ & \nwarrow \swarrow_{(A,F_A)} }$

So I guess we’d have to replace here now $\Omega^\bullet(X)$ with the weak Lie 2-algebroid we found above to get the Dirac operator

$\array{ & \swarrow \nwarrow^0 \\ \Omega^\bullet(X) &\Downarrow^{(e,D e)}& \mathrm{inn}(g)^* \\ & \nwarrow \swarrow_{(A,F_A)} }$

Or something like that.

When this is undersood for $g = \mathrm{spin}(n)$ we should repeat it for $g = \mathrm{string}(n)$ .

Posted by: Urs Schreiber on October 13, 2007 4:44 PM | Permalink | Reply to this

Re: Categorified Clifford Algebra and weak Lie n-Algebras

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I should add that I expect there even further generalizations to be found:

not only should we be looking for the differential algebras dual to fully weak Lie $n$ -groupoids.

I expect that also finite (non-Lie) $n$ -groupoids are somehow Koszul dual to certain differential algebras.

This is what the considerations suggest that I looked at together with Eric Forgy:

to every directed graph with the property that between anyy two vertices there are edges in only one direction, we can canonically associate a differential algebra which plays the role of a discrete approximations to the deRham complex, with the discreteness modeled by the graph.

This differential algebra differs from the differential algebras that are Koszul dual to Lie groupoids by the fact that it is non graded commutative (above degree 0) – but the failure of graded commutativity is well behaved. It precisely measure the finiteness of the graph.

Eric and I were always trying to identitfy the right categorical structure underlying this construction. While there are a couple of obvious guesses, it never became quite clear to me.

Now I am thinking: certainly we need to be looking into the same kind of duality that related Lie groupoids with differential graded-commutative algebra.

I need to think more about this. Does anyone have any idea about this? Is there anything known about generalizing Koszul-duality of Lie $n$ -groupoids to the non-Lie (“discrete”, “finite”) case?

The big advantage of the differential algebra associated to a graph, as Eric emphasized so much, is that it gives a beautifully hands-on finite model for (Riemannian) differential geometry. It would be nice if we could obtain a precise statement that relates this to something like Koszu-duality with finite grouoids. That should help understand the generalization in the other directions – Grassmann to Clifford in particular – more easily.

The following table lists the relation between groupoids and Koszul-dual (or whatever the right term turns out to be) differential algebra, the way it is known and the way that I am conjecturing it here. This should be just a small part of a larger such table:

$\array{ \mathbf{groupoids} && \mathbf{differential algebra} \\ Lie, weak assoc., strict inv., abelian && Grassmann \\ Lie, weak assoc., strict inv., nonabelian && differential Grassmann \\ Lie, weak assoc., weak inv., abelian && Clifford \\ Lie, weak assoc., weak inv. && differential Clifford \\ non-Lie, strict inv. && differential non-commutative (above degree 0) algebra }$

Posted by: Urs Schreiber on October 14, 2007 4:55 PM | Permalink | Reply to this

Re: Categorified Clifford Algebra and weak Lie n-Algebras

Mmm… interesting!

Posted by: Bruce Bartlett on October 14, 2007 5:30 PM | Permalink | Reply to this

Re: Categorified Clifford Algebra and weak Lie n-Algebras

I’m surprised I missed this!

Better late than never. I’m currently putting some effort into understanding this, starting with some questions here.

Posted by: Eric Forgy on September 25, 2009 6:40 PM | Permalink | Reply to this

Read the post Loday and Pirashvili on Lie 2-Algebras (secretly)
Weblog: The n-Category Café
Excerpt: On the non-standard monoidal structure on 2-term chain complexes induced from the monoidal structure on Baez-Crans 2.vector spaces.
Tracked: October 16, 2007 5:05 PM

Read the post On Lie N-tegration and Rational Homotopy Theory
Weblog: The n-Category Café
Excerpt: On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan's old article on this issue in particular.
Tracked: October 20, 2007 5:18 PM

Re: Categorified Clifford Algebra and weak Lie n-Algebras

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I talked to a couple of people about this idea that we should be looking at dg algebras whose underlying alsgebra is a graded commutative one modulo some Clifford-like relations.

To avoid misunderstandings, I formulate this more precisely:

For $V$ a graded vector space, and $k$ a graded-symmetric bilinear grade-preserving map on it of the form $k V \otimes V \to V \,,$ I’ll say that the graded Clifford algebra corresponding to this is the graded symmetric algebra

$S(V)$

modulo all relations

$v w - (-1)^{-1} w v = k(v, w) \,.$

Then a differential graded Clifford algebra is a graded Clifford algebra with a degree +1 differentiation on it, that squares to zero.

Notice that ordinary Clifford algebra on an ungraded vector space $(V,\langle \cdot,\cdot\rangle)$ is subsumed in this definition by regarding $V$ to be in degree 1, adding a single degree 2 generator $b$ and setting $k(v,w) = \langle v,w\rangle b \,.$

As I said before, an ordinary Clifford algebra would be, from this point of view, a skeletal Lie 2-algebra with an ordinary Lie algebra of objects and with the symmetrizator given by the bilinear form.

Posted by: Urs Schreiber on October 28, 2007 1:22 AM | Permalink | Reply to this

Read the post Higher Clifford Algebras
Weblog: The n-Category Café
Excerpt: A talk by Chris Douglas reporting on his work with Arthur Bartels and Andre Henriques on "higher Clifford algebras". They're related to elliptic cohomology and they form a 3-category!
Tracked: October 30, 2007 12:39 AM

Read the post Differential Graded Clifford Algebra
Weblog: The n-Category Café
Excerpt: On Clifford-defomrations of Chevalley-Eilenberg and Weil algebras.
Tracked: August 17, 2008 12:40 PM

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October 9, 2007