### Landsman on Quantization of Poisson Algebras Associated to Lie Algebroids

#### Posted by Urs Schreiber

In the last entry I mentioned the work in

N.P. Landsman, B. Ramazan
*Quantization of Poisson algebras associated to Lie algebroids*

arXiv:math-ph/0001005

summarizing it with the remarkable slogan

Quantization is Lie integration.

true at least in all the cases where the classical Poisson manifold to be quantized is the dual of a Lie algebroid or of a coadjoint orbit inside of that.

This article states and proves a theorem (two versions, actually) which asserts that the $C *$-algebraic deformation quantization of such a Poisson manifold is the groupoid C-star algebra of one of the Lie groupoids integrating the given Lie algebroid.

**$C^*$-algebraic deformation quantization**

As nicely reviewed in the introduction, there are several ways to make precise what deformation quantization is supposed to be.

In some way or other, one wants to start with a Poisson manifold $X$ (maybe assuming for simplicity that it is actually symplectic) and find a non-commutative deformation of the commutative algebra of functions on $X$ such that the commutator in the deformed algfebra is “approximated to first order” by the original Poisson bracket.

One way to approach this is by considering algebras in formal power series of a single formal parameter $\hbar$. That “formal” approach was famously shown by Kontsevich to always admit a solution to the deformation problem, we discussed that in That shift in dimension.

But in a sense this solution is unsatisfactory precisely because it is “formal”: there is in general no guarantee that one can replace $\hbar$ by an actual number (in particular by the number 1!) and get converging expressions. But that is need to really have a “physical quantization” in the ordinary sense.

Landsman and Ramazan review a different approach to the deformation quantization, one which does not suffer from this problem: they define a deformation of the Poisson algebra as “continuous field of C-star algebras” indexed by $\hbar \in [0,1]$ which interpolates between the Poisson algebra and the noncommutative C-star algebra at $\hbar = 1$ quantizing it.

**Poisson manifolds from duals of Lie algebroids**

Lots of relevant Poisson manifolds turn out to be duals of Lie algebroids, with the Poisson structure being essentially nothing but the Lie bracket on the Lie algebroid. This is best seen from the archetypical example, the canonical Poisson manifold $T^* X$, which is the dual to the tangent Lie algebroid $T X$.

In

N. P. Landsman
*Strict deformation quantization of a particle in external gravitational and Yang-Mills fields*

Journal of Geometry and Physics, Volume 12, Issue 2, p. 93-132.

this example is generalized to the situation of the general *charged particle*, in which case the Lie algebroid in question is the Atiyah Lie algebroid of the principal bundle giving the background field – and the quantum algebra is the groupoid algebra of the *transport groupoid* aka *gauge groupoid* of this bundle.
(One has to think of this algebra as acting on sections of this bundle regarded as equivariant functions on the total space of the bundle. More on that later.)

**Coadjoint orbits**

In the special case that the bundle in question lives over a point, so that its total space is nothing but the structure group itself, the Lie algebroid in question is just a Lie algebra and the corresponding quantum algebra is the convolution $C^*$-algebra of the group – the continuous analog of the group algebra of a finite group.

In this case it is well known that the Poisson manifold $g^*$ is foliated by other Poisson manifolds: the coadjoint orbits. Their quantization corresponds to irreducible parts of the quantization of the full thing. There is a big interesting theory about coadjoint orbits.

I notice that one nice way to think of a coadjoint orbit in a Lie algebra is as the orbit in its Chevalley-Eilenberg DGCA of a homogeneous element under the *inner automorphism group* along the lines described here:

the orbit is of the form

$exp([d,\iota_v]) t$ for t in $g^*$, $v$ any Lie algebra element, $\iota_v$ the derivation on $CE(g)$ obtained by contracting with it and $d$ the differential on $CE(g)$.

This immediately generalizes to a notion of coadjoint orbits on arbitrary Lie algebroids and in fact Lie $n$-algebroids.

I haven’t checked but would guess that Landsman’s discussion should generalize the coadjoint orbits of arbitrary Lie $n$-algebroids. That should give a huge source of examples. So I am led to wonder if the slogan at the beginning of this post actually holds in complete generality.

[more later]

## Re: Landsman on Quantization of Poisson Algebras Associated to Lie algebroids

One thing about Landsman and Ramazan’s approach and also that of Eli Hawkins is that I don’t see where the idea of a

polarizationenters. Landsman and Ramazan’s introduction gives a wonderful summary of all things quantum; I love the lines(this makes it seem like a criminal offence!) and also

Anyhow, in their review of geometric quantization, they look at

prequantization, discover a flaw in the naive approach, and then conclude that one must adopt deformation quantization. Isn’t this a straw man argument?