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Let's have some fun with the Heisenberg group H! It is the group of upper triangular matrices of the form

/ 1 a c \
| 0 1 b |
\ 0 0 1 /

Muliplying two such matrices you can easily see it's a monoid, because the result is again an upper triangular matrix:

/ 1 a+A c+aB+C \
| 0 1 b+B |
\ 0 0 0 /

1/

You can also read this thread on my blog:
blog.deinding.net/maths/heisen

· · Web · 1 · 6 · 9

Of course, the identity matrix Id is also the identity of the monoid. This also makes clear that the group is non-commutative. The inverse is also rather easy to find, and that makes it a group:

/ 1 -a ab-c \
| 0 1 -b |
\ 0 0 1 /

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Two ways to embed an affine transform:

If we let such a matrix act on 2-vectors embedded like this

/ x \
| y |
\ 0 /

it gives an affine transform, that is, a shear plus a translation:

/ 1 a \ / x \ + / c \
\ 0 1 / \ y / \ b /

Notice that there are two ways such a 2-dimensional affine transform can be embedded in the Heisenberg group!

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Lie algebra:

The way I described it, you're probably thinking of real numbers for a, b, and c, and that makes it a continuous- or Lie group! Let's stick with that (the discrete Heisenberg groups are also fun to explore, have a look on Wikipedia's take for some more on this).

/4

As such, it is nilpotent, and non-abelian, which I had mentioned above already. There is also an associated Lie algebra, which I find to be a nice example to illustrate the relation between Lie groups and Lie algebras. I think so, because it's small, and because the exponential map, which always translates a Lie algebra to a Lie group, is bijective in this case! Which is quite obvious if you look at it:

5/

/ 0 a c \ / 1 a ab+c \
exp | 0 0 b | = | 0 1 b |
\ 0 0 0 / \ 0 0 1 /

There's a fun way to generalize all this to higher dimensions using a row vector a a column vector b and a scalar c:

H_{2n+1} =

/ 1 a c \
| 0 Id_n b |
\ 0 0 1 /

This almost looks just like what I wrote earlier, but with the types of some entries changed! All the other things also works out just as nicely!

6/

Polarized Heisenberg group

If you look at this the right way, you get an even general picture of the Heisenberg group, for symplectic space!

Remember, symplectic space is this freak 2n-dimensional space where, instead of a dot product, we get a product which yields zero when its argument vectors are parallel! (This is then called a nondegenerate skew-symmetric bilinear form). Now, if we pick a Darboux-basis, we can get the corresponding Heisenberg group to look almost like before again!

7/

Picking a basis for a symplectic space V amounts to choosing a Lagrangian, and that is sometimes often called the polarization of V, which leads to the beautiful name polarized Heisenberg group for a representation of this form. For some more details on this see here:

Heisenberg Group on Wikipedia - en.wikipedia.org/wiki/Heisenbe

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Nilgeometry

William Thurston solved the geometrization conjecture for 3-space. That is, he found eight homogeneous geometries (all points look the same) for 3-space, in analogy to spherical, euclidean, and hyperbolic geometry in two dimensions. One of these is called Nil.

9/

Thurston's space sort of looks like a plane, but if you move in a circle, you also move perpendicular to the plane! It is similar to an example for an effect called holonomy: when you roll a ball in a small circle on a plane, the ball ends up twisted along an axis pointing away from the plane. While moving in two dimensions on a plane, we have also moved using a third degree of freedom. The difference is that in nilgeometry that third direction does not repeat every full turn.

10/

The notion of Nilgeometry was introduced by Anatoly Mal'cev in 1951 as the quotient N/H of a nilpotent Lie group N modulo a closed subgroup H. Just so you know where the funny name comes from. For solvable Lie groups there is an analog, and there's a corresponding Thurston geometry beautifully named Sol!

11/

In any case, there's a representation of the Nilmanifold which, again, looks almost like the things we did earlier! Of course these ideas make sense in dimensions other than three, see here for a more complete account:

Nil manifold on Wikipedia - en.wikipedia.org/wiki/Nil_mani

Nilgeometry explained - youtube.com/watch?v=FNX1rZotjj

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The Heisenberg group turns up in quantum mechanics.

> [In] Weyl quantization, proposed by Hermann Weyl, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. - en.wikipedia.org/wiki/Quantiza

13/

We can fix generators for the small Heisenberg group like this:

/ 1 1 0 \
x = | 0 1 0 |
\ 0 0 1 /

/ 1 0 0 \
y = | 0 1 1 |
\ 0 0 1 /

That leads to the center generated by z = xyx^-1y^-1:

/ 1 0 1 \
z = | 0 1 0 |
\ 0 0 1 /

14/

If that doesn't mean much to you, at least notice that the center is the part of a group which is commuative. Look what happens if we substitute the identity for z and right-multiply first by y and then by x:

Id = xyx^-1y^-1
yx = xy

15/

I secretly wanted to make this post about Weyl algebra, also called symplectic Clifford algebra, another name I find irresistable. Let me say just this much: Herrmann Weyl wanted to understand why the Schrödinger picture of quantum mechanics is equivalent to Heisenberg's, and that's why he came up with this stuff.

This post is already too long, so I will stop here.

Thanks to ZenoRogue for showing me the relation between Nil-geometry and the Heisenberg group, which got me started!

16/

Links:

Here's a seminar with two speakers talking about the Heisenberg group: History for Physics - "The Weyl-Heisenberg group: from quantum mechanics to quantum information" - youtube.com/watch?v=atgyvRvS1I

The first part delves into a specific problem set, the second half gives a more bird's eye account on the history of Heisenberg group.

Edward Teller "Understanding Group theory with Heisenberg" - youtube.com/watch?v=2OiRL6DfFw

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Mostly, my variable width ascii art came out okay'ish, but here it failed badly. I suppose it's still kind of readable.

Anyways, here's a photo of that bit from my blog's LaTeX rendering. I know, mathstodon.xyz can render LaTeX as well, but I didn't like the look of matrices on other clients, so...

Maybe I should simply take pictures right away next time.

@RefurioAnachro - great thread. "William Thurston solved the geometrization conjecture for 3-space" is not quite right. He posed the conjecture, but it was proved by Perelman:

en.wikipedia.org/wiki/Geometri

The Poincaré conjecture can be shown as a corollary!

Thank you! It seems I do not understand what the geometrization conjecture is! I thought it was about finding homogenous geometries and that it was solved by Perelman for d>3.

So then what was it that Thurston proved by showing that there are 8 homogeneous geometries for d=3? Is there also a catchy name for it? I seem to have run into a very similar mistake a couple of weeks ago, to be corrected by @zenorogue. That's what you get if I don't blog about my finds right away :/

@johncarlosbaez

@RefurioAnachro - the geometrization conjecture is explained in that Wikipedia article I linked to. Roughly, it says that every compact 3-manifold can be decomposed in a canonical way into pieces that each have one of 8 types of homogeneous geometrical structure.

So this conjecture goes way beyond proving that there are 8 homogeneous geometries in 3d. That was done by Thurston. That's basically just a matter of Lie group theory - not so hard.

(He also did harder stuff.)

@johncarlosbaez @RefurioAnachro Also "there are 8 homogenoeous geometries in 3d" is misleading IMO -- there are 8 geometries useful in the context of geometrization theorem (classifying closed manifolds), but there are also homogeneous complete simple-connected manifolds which are not useful there, because they are less symmetric variants of Thurston ones (Berger sphere) or because there are no closed manifolds with a given geometry.

Cool, so there are more than 8 homogeneous, simply-connected 3-geometries! I'm not sure why I should care about their completeness, but Berger's sphere is complete. Here

en.wikipedia.org/wiki/Berger%2

they say its metric is \(βω_1²+ω_2²+ω_3²\) which looks just like a mere sphere's except for that β. And something about dual 1-forms I didn't get. So it's not like a 3-analog to the surface of an ellipsoid?

Wait, β acts along a Hopf fibration? I can't quite see it.

@zenorogue @johncarlosbaez

Until now my thinking has been that the 8 Thurston geometries are the 3d analog for the tree planar (spherical, euclidean, and hyperbolic) geometries. Have I been mislead there, too? We can put a metric on the projective plane, but it's simply hyperbolic, right? Basically all 2-manifolds are hyperbolic, but that happens, because the complex plane is so powerful. And there's no equally powerful pendant in d=3.

@zenorogue @johncarlosbaez

@RefurioAnachro @johncarlosbaez In 2D there are three homogeneous geometries because they differ only by curvature which is a single number (assuming we consider K=1 and K=2 the same because they differ only by scale). Not sure why you think projective plane is hyperbolic. By identifying antipodal pairs on S2, you get elliptic plane which has this topology, and spherical geometry.

@RefurioAnachro @johncarlosbaez I do not know much about the classification of 3D Lie groups but if you look at Solv, it is easy to see that you can also get a different geometry by changing a parameter (like in arxiv.org/abs/2005.06430 ). For some values of alpha you get Thurston geometries (H2xR, H3 and Solv) and for others you get geometries that are not Thurston because there are no closed manifolds with these geometries.

@RefurioAnachro @johncarlosbaez Nil is a "twisted product" of E2xR, in the sense that a loop in E2 moves you along the xR fiber by proportional to the area of the loop. Twisted H2xR also appears in Thurston geometries as ~SL(2,R). The lack of twisted S2xR may be surprising. This is because you simply get S3 (with Hopf fibers).

@RefurioAnachro @johncarlosbaez ... But only if the if the proportionality constant is 1. Otherwise you get the less symmetric Berger sphere. Anything with Berger sphere geometry can be also given S3 geometry, so it is useless in the context of Thurston geometrization. (In case of twisted H2xR you can also change the proportionality constant.) I have some visualizations here: twitter.com/ZenoRogue/status/1 (we did not know the name Berger sphere then)

Those are some awesome videos, @zenorogue! I may have looked at them then, not realizing that these are really something different from the other twisted geometries!

Your explanation for the sphere made Berger's sphere click in place for me! Thank you!

@johncarlosbaez mentioned that the classification of Lie groups is not too difficult. That is, if you can handle Lie algebras conceptually. Like Coxeter-Dynkin diagrams. Which I ought...

I guess we can interpolate between root systems. I'm not sure if Coxeter-Dynkin diagrams themselves are a good way to sketch this. Or perhaps they are, labeling vertices of a Cayley graph. A Lie group (err... -oid?) made of Lie groups?

All this is tantalizing. Obviously, I don't know enough about these things. I have some rough pictures in my mind, but I really should spend some time writing about it!

@RefurioAnachro @johncarlosbaez Regarding completeness -- we want a natural representative of the class of all manifolds with the given geometry. So we take the simply-connected complete one (so for example E^3, not some random simply-connected open subset of E^3, that would not be complete).

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