The n-Category Café

Before coming to a couple of interesting facts, a

Quick reminder of some concepts

Given a manifold $X$ and a bivector $$\pi \in \Lambda^2 T X$$ we obtain an antisymmetric product on functions on $X$ given by

$$\pi : C^\infty(X) \otimes C^\infty(X) \to C^\infty(X)$$ $$(f,g) \mapsto \{f,g\} := (\pi(f)g) \,.$$

So if in a local chart we have $$\pi = \pi^{ij} \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j}$$ then $$\{f,g\} = \pi^{ij} (\frac{\partial}{\partial x^i} f) ( \frac{\partial}{\partial x^j} g) \,.$$

The tensor $\pi$ is called a Poisson tensor if this bracket is a Lie bracket, i.e. if it satisfies the Jacobi identity. A manifold equipped with a Poisson tensor is a Poisson manifold.

In the game called “geometric quantization” one wants to stat with a Poisson manifold and constrcuct from it in a natural way a Hilbert space and/or the algebra of bounded operators on that Hilbert space.

Moreover, one wants to send the commutative algebra of functions on $X$ to some usually non-commutative sub-algebra of that big algebra in such a way that the commutator in this non-commutative algebra is “approximated” by the Poisson bracket.

The following example seems to be very fundamental and important, but I wasn’t really aware of it before. Now I am.

Geometric quantization of the dual of a Lie algebra.

For $g$ a Lie algebra, the dual vector space $g^*$ is naturally equipped with a Poisson structure:

first consider linear functions $f_1, f_2 : g^* \to \mathbb{R}$ on $g^*$. These are nothing but elements of $g$. So a natural bracket operation on them is the Lie bracket. This extends, apparently, uniquely to a Poisson bracket on all functions on $g^*$.

Let $(g^*, \pi_{[\cdot,\cdot]})$ be $g^*$ regarded as a Poisson manifold with this particular Poisson structure. Then we can try to apply geometric quantization, or some generalization of it which also applies to Poisson manifolds. Let $C_0(g^*)$ be a space of functions on $g^*$ that go to 0 at infinity in a sufficiently nice way. The quantization process should send this commutative algebra to some possibly non-commutative algebra.

$$C_0(g^*) \stackrel{quantization}{\mapsto} A \,.$$

What is that algebra? Apparently, turning some crank or other and making some non-canonical choices which one has to make in this quantization business, that algebra turns out to be the convolution algebra of one of the Lie groups $G$ integrating $g$ (the choices corresponding to which Lie group one gets)

$$C_0(g^*) \stackrel{quantization}{\mapsto} C^*(G) \,.$$

This convolution $C^*$-algebra is the Lie analog of the group algebra of a finite group.

Eli did not explain many of the details going into this quantization of algebras. He says that this statement can be found in the book

N. P. (Klaas) Landsman
Mathematical topics between classical and quantum mechanics

which I haven’t seen yet. But details are in

N.P. Landsman, B. Ramazan
Quantization of Poisson algebras associated to Lie algebroids
arXiv:math-ph/0001005 .

In there the statement is actually generalized to Lie algebroids:

Geometric quantization of the dual of a Lie algebroid.

The fiberwise dual $A^*$ of the vector bundle $$\array{ A &\to& T X \\ & \searrow \swarrow \\ & X }$$ underlying a Lie algebroid $A$ also inherits a Poisson structure $\pi_{[\cdot,\cdot]}$ from the bracket on sections of $A$ given by the Lie algebroid structure of $A$.

So we can ask what the quantization of the Poisson manifold $(A^*, \pi_{[\cdot, \cdot]})$ is. As you can now guess if you meditate over the above statement for a second is that the quantized algebra of functions $C_0(A^*)$ on $A^*$

$$C_0(A^*) \stackrel{quantization}{\mapsto} C^*(Gr) \,,$$

is the $C^*$-convolution algebra of $Gr$, where $Gr$ is one of the Lie groupoids integrating the Lie algebroid $A$.

Crucial example

In particular, take $A = T X$ to be the tangent Lie algebroid of some manifold $X$. Then the dual space $A ^* = T^* X$ is the cotangent bundle.

The Poisson structure induced on that cotangent bundle by the above general procedure is just what you’d expect: it’s the actually symplectic structure of the symplectic 2-form $d \alpha$, where $\alpha$ is the canonical 1-form on $T^* X$.

We already know what the quantization of this Poisson manifold $T^* X$ should be. After all, this is the example that got the business of quantization started in the 1930s! The algebra we want to see come out is that of operators on square integrable functions on $X$.

And indeed, by the above statement we find that the algebra of functions on $T^* X$ is sent to the convolution algebra of the groupoid integrating the tangent Lie algebra – which is the fundamental groupoid of $X$ or one of its quotients, such as the pair groupoid of $X$:

$$C_0(T^* X) \stackrel{quantization}{\mapsto} C^*(PairGroupoid(X))$$

but that indeed is $K(L^2(X))$ – compact operators on the Hilbert space of square integrable functions on $X$, in a direct generalization of how the groupoid algebra of the pair groupoid of any finite set of cardinality $n$ is that of $n \times n$ matrices.

This deserves a slogan.

This is too good to remain unnoticed, so it deserves a catchy slogan. Maybe this one:

Quantization is nothing but Lie integration.

Hm…

If you allow me to add links to the above slogan, then it reads

Quantization is nothing but Lie $n$-tegration.

Well, in the above context this is true for Poisson manifolds which are total spaces of Lie algebroids. So this cannot be true generally without further modification. But it’s still suggestive.

Bisections and inner automorphisms

Eli Hawkins mentions that from a Lie groupoid $Gr$ over $X$ one obtains the Lie algebroid differentiating it (or rather, directly, it’s Lie-Rinehart pair, I would say) by taking the Lie algebra of the group of bisections.

A bisection here is define to be a section of the source map of the Lie groupoid which is also a section of the target map. Bisections have an obvious notion of composition which makes them into a group.

In more categorical terms this means, in other words, that a bisection is an arrow field on the groupoid which is the component map of a natural transformation $\eta$ which related the identity automorphism of the groupoid to some other automorphism $\alpha$.

$$\array{ & \nearrow \searrow^{Id} \\ Gr &\Downarrow^\eta& Gr \\ & \searrow \nearrow_{\alpha} } \,.$$

Drawn this way, it is clear which group these bisections sit in: the inner automorphism 2-group of $Gr$.

In fact, we have talked about this way of getting Lie algebroids from differentiating the inner automorphism groups of their Lie groupoids a lot here, in the general context of “Arrow-theoretic differential theory”.

For instance the discussion of the fact that the Lie algebroid of $Gr$ comes from the Lie algebra $Lie(INN(Gr))$ is discussed in section 4.1 of Tangent categories, and also on Supercategories.