|Oct12-06, 04:15 AM||#1|
Re: Klein's Definition of Geometry
On Thu, 18 Aug 2005, Perspicacious wrote:
> Is there an introductory geometry textbook that explains how to
> interpret the basics of an abstract geometry from a specific
> transformation group?
Alas, Klein himself never properly wrote up his ideas on "Kleinian
geometry", an omission which continues to have ill consequences in our own
day. He -did- write some classic books with much more limited aims, which
do scratch some of the surface of this subject, but he never wrote the
requisite masterpiece fully explaining the background, principles and main
examples known in his day of the Erlangen program itself, with the result
that today his ideas tend to be underappreciated. These ideas have
continued to gather importance with the passage of time, but unfortunately
developments have spread into areas increasingly isolated by artificial
barriers of terminology and notation which make it difficult for
researchers (must less students) to grasp the big picture, or even to
assess the extent of Klein's influence on the shape of modern mathematics.
Sadly, there is -still- nothing I could call a proper textbook on Kleinian
geometry, nor even (to my knowledge) an adequate survey article of the
modern mathematical landscape from the perspective of Kleinian geometry.
The best I can do is to
1. mention some resources which offer a few hints,
2. sketch partial answers to some of your subsequent questions.
Many of the most interesting examples of Kleinian geometry at work involve
either matrix Lie groups (especially the "classical groups") or finite
permutation groups. To grasp Klein's vision you will need to know
something about group actions, Lie groups and Lie algebras, permutation
groups, classical transformation geometry, Lie's theory of differential
equations, and invariant theory.
For the interaction of Lie theory (Lie groups and Lie algebras) with
Kleinian geometry, I recommend you start with two expository articles
Richard Millman, "Kleinian transformation geometry", American Mathematical
Monthly 84 (1977): 338-349.
Roger Howe, "Very basic Lie theory", American Mathematical Monthly 90
These are not error-free (see the erratum in the next issue after Howe's
article; no-one bothered to point out errors in the earlier article but
they are alas there), but AFAIK they are the best brief accounts currently
available. One of the essential notions you may appreciate after reading
the article by Millman is the notion of a hierarchy of structure in
geometry, which is one of the most important and characteristic features
of Kleinian geometry.
The classic textbook Geometry by Coxeter should give you a fair grasp of
"synthetic" techniques in "transformation geometry", a modern name for a
portion of Kleinian geometry which is sometimes taught to future teachers
of high school geometry to deepen their appreciation of the beauties of
even "elementary euclidean geometry" of the plane. This should help you
get used to thinking about matrix Lie groups of transformations and their
generators, which is a key ingredient of Kleinian geometry, and may help
you grasp some of the beautiful correspondence between certain subgroups
of a symmetry group and various "geometric elements" (e.g. points, lines
and planes) in the corresponding geometry. However, you really need to
master the relationship of these synthetic techniques with certain
analytic and algebraic techniques; some of the books I mention below might
At a higher level I highly recommend at least the first two parts of the
book Lectures of Lie Groups and Lie Algebras by Carter, Segal and
MacDonald. This lovely book is a bit sketchy but gives a marvelous
overview of a huge, huge subject.
The book Indra's Pearls might be worth a look, although I haven't yet seen
it myself and am not quite sure if it is a popular book or not, or whether
it is about "Kleinian geometry" in the sense I think you mean or "Kleinian
groups". But I know John Baez thinks highly of this book, and (at his
website) you can look at Week 178 and the five or six preceding and
suceeding weeks for a readable explanation of some beautiful material on
reflection groups which should help you to understand the concept of
While Klein and Lie circa 1870 were most concerned with matrix Lie groups,
there are also geometries (such as "finite projective planes") which have
finitely many points and lines. These have symmetry groups too, so
Klein's ideas apply in these situations as well.
An excellent textbook which studies group actions by both matrix Lie
groups and finite permutation groups is Neumann, Stoy, and Thompson,
Groups and Geometry. Unfortunately right now I can't think of a really
adequate presentation of finite projective planes and the groups GL(n,q),
PGL(n,q) and so forth, but you certainly will need to know the basic facts
about these, and this book does say a tiny bit about finite projective
planes. The textbook Permutation groups, by Peter J. Cameron, is also a
bit sketchy but complements the book by Neumann et al. very well is a
great place to learn more about finite simple groups of Lie type (one of
the really key examples for Kleinian geometry). These might help you put
the stuff discussed by Coxeter in a broader context.
It turns out to be extremely instructive to study finite groups and Matrix
Lie groups side by side. Not only do you gain insight about one from the
other, but it turns out that there are beautiful connections between these
two worlds. One of the nicest ways to begin to see why is to go to John
Baez's website and read Week 178 and the five or six preceding and
suceeding weeks, which concern parabolic subgroups of reflection groups
and some lovely connections with combinatorics, topology, and invariants.
Invariant theory is an -absolutely neccesary- for understanding Klein's
ideas. This beautiful subject was a standard part of mathematical
education in the 19th century but tragically was dropped by about 1940,
despite the fact that the importance of the subject has not lessened one
iota (if anything, its importance has -increased-). It would be good to
know something about invariants for both matrix Lie groups and their
finite cousins, but the former is truly essential.
I recommend starting with the classic textbook Ideals, Varieties, and
Algorithms by Cox, Little, and O'Shea, and then studying chapters one and
two of the beautiful little book Algorithms in Invariant Theory by Bernd
Sturmfels. The former book is also by far the best extant textbook on
elementary algebraic geometry, another classic subject which was standard
in Klein's day but has since been dropped from the UG and even first year
graduate curriculum, even though its importance has if anything
-increased- in both mathematics and its applications. This material will
also be invaluable in understanding the context of the work of Klein and
Lie around 1870.
The textbook Representations and Characters of Groups by James and Liebeck
will be an excellent companion to this study, since today many ideas which
in 1870 would have been considered in terms of permutation representations
are more likely today to be considered in terms of linear representations.
There is by no means a dictionary between these worlds, so the emphasis
has changed in ways serious students need to understand. Then you can
read Richard Kane's book Reflection Groups and Invariant Theory for still
more lovely material closely connected to the stuff John is talking about.
I can't overemphasize the degree to which reflection groups and Weyl
groups provide a particularly apt source of examples for Klein's ideas.
Going back to the interaction of Lie theory, differential geometry, and
Kleinian geometry, the idiosyncratic textbook Differential Geometry by
Richard Sharpe does offer an introduction to a small portion of Kleinian
geometry and its generalization to Cartanian geometry. Unfortunately
Sharpe gets bogged down in technical isues and never gets to the fun
stuff; he only treats a few limited aspects and unfortunately never
mentions the applications of the stuff he does cover to gauge theories in
physics. This book probably won't be much help by itself, but with a
sufficient amount of supplementary material from other sources it should
be valuable. It is the only differential geometry book I know of which
even -attempts- to provide an introduction to even a tiny piece of
Kleinian geometry under that name.
The book Equivalence, Invariants, and Symmetry (among others) by Peter J.
Olver proably offers more useful hints, depending upon the motivation for
your interest. These books will be invaluable in learning enough about
Lie's theory of differential equations to understand the origins of the
subject in the mathematical conversations of Lie and Klein in roughly the
For classic examples of Kleinian geometries and an important duality
(discovered by Klein) between "angle" and "distance", some old
undergraduate level books by Yaglom (also transliterated Iaglom) are still
There are also several books on the theory of buildings, but again these
will probably be hard to follow for anyone lacking the tacitly assumed
background, and even worse, it might be hard to see the point unless you
already know a fair amount about Klein's vision.
This reading list must appear very daunting, which it is really
unfortunate, since I have done all this reading (and much more), and am
confident that with a great deal of work by a sufficiently industrious
author, it -is- possible to pull all the really essential stuff together
into a good book giving a survey of the lay of the land, which would no
doubt encourage many more advanced students to try to explore further.
Several related books (as in Peter Olver's corpus) would be even better.
So I recognize that there is an urgent need for a proper book one can cite
rather than citing a list of books and urging a student to somehow pull a
huge amount of material together on his own. If I dared, I would
apologize on behalf of the entire mathematical profession for the
lamentable lack of an adequate textbook surveying the scope and
signficance of Klein's ideas :-/
> Is carrying out Klein's Erlanger Program straightforward or elusive?
The answer to your question depends upon what you mean by "Klein's
program". I apologize in advance for a neccessarily very sketchy and
opague description of some of the issues which arise when you look into
Many people think of the Erlangen program as running something like this:
given a geometry, find its symmetry group (the transformation group
preserving all the geometrical relations making sense in this geometry),
describe the polynomial and differential invariants of this group, and
study the geometric interpretation of these invariants.
Here, simply -finding- the symmetry group is often straightforward, but
finding a useful description of the relevant invariant rings can be
anything but straightforward, notwithstanding the fact that huge
theoretical progress in the theory of invariants was made in Klein's day,
especially by his colleague Hilbert, and much more since then, especially
in computer algebra techniques for computing with invariant rings. Many
hard problems remain, but one reason why I stress the role of reflection
groups (and finite permutation groups) as a source of examples is that
here we have effective algorithms, and it is in situations where you have
a convenient algebraic description of the invariants that the power of
Klein's approach really becomes clear, because you can follow how his
algebraic/geometric correspondence works in great detail in concrete
In short, depending upon what group and what kind of invariants you are
studying, this direction may not be entirely "straightforward".
Klein was also interested in going in the other direction: given a group
G, construct a reasonable "geometry" for which G is the symmetry group.
In particular, around 1870, when Klein and Lie were working very closely
together, Lie was very interested in figuring out what kind of "geometry"
corresponds to the symmetry group of an ordinary differential equation.
It sounds like this might be the direction you are interested in. If so,
it probably matters whether you are interested in the "geometry" arising
from something like a classical matrix group (e.g. for typical
differential equations of mathematical physics like the Laplace equation),
or a finite permutation group. Let me say a bit first about classical
groups and differential equations (the context in which Klein's ideas
first arose in his work with Lie).
In Klein's day, many "noneuclidean geometries" had been introduced and a
general picture was known, e.g. the rough picture for the most commonly
encountered plane geometries is a partial order (left to right) suggested
by this ASCII diagram
projective -- affine congruence geometry
The importance of Klein's work was that he tied this up with -algebraic
features- of the corresponding symmetry groups and their invariant rings
in a fairly precise way, thus unifying most of the geometries known in his
day into a coherent "algebraicogeometric picture".
Indeed, especially in higher dimensions there are more notions of
geometry, particularly "symplectic geometry", which turn out to fit very
neatly into the classical group picture from the perspective of Kleinian
geometry. Symplectic geometry turns out be essential for important parts
of dynamical systems (we are back to differential equations here!) as well
as many parts of physics.
A few remarks about the sketch diagram of the hierarchy of plane
geometries might help convey the flavor. Projective geometry has no two
point invariants at all, but it has a four point invariant (cross-ratio),
and the three point relation "colinear" and the six point relation
"coconic" (and so forth, for higher degree plane curves). Affine geometry
has a notion of "convex hull", but it has no notion of area or angle.
Similarity geometry has a notion of "angle", but no notion of distance.
Areal geometry has a notion of "area" (e.g. of a triangle), but no notion
of angle. Congruence geometry has a notion of "distance", and inherits the
notions of angle and area from the less rigid similarity and areal
geometry. Actually, there is more than one notion of distance and angle,
but the euclidean and Minkowski geometries share the same notion of area.
Following up a remark of Cayley, Klein found a beautiful projective
duality between angle and distance, which helps to explain why these two
notions are so closely tied up with each other.
Note that "congruence geometry" has a notion of "congruent triangles" and
a less rigid notion, "similar triangles". Only the latter makes sense in
its parent geometry, "similarity geometry". Here, a relation makes sense
in a geometry precisely in case it is preserved by the symmetry group of
that geometry, and in this case, it corresponds to an appropriate
By the way, symplectic geometry, like euclidean and pseudoeuclidean
congruence geometry, is defined using an appropriate quadratic form, so it
is very close to the classic origins in algebraic geometry and invariant
theory of Klein's ideas. See the article by Howe and the book by Segal et
al. for more information.
We can of course extend this picture leftwards to include much more
general transformation groups, say the groups of diffeomorphisms and then
homeomorphisms of the plane, etc. However, to its central importance in
algebraic geometry, Klein and his colleague Hilbert mostly tended to
regard projective geometry as the mother of all geometries.
Hilbert found beautiful and very detailed relations between geometric
properties of projective geometries (and its more rigid children) and what
we'd now call "free resolutions". It turns out that by homological
algebra you can tell whether or not three points are colinear, coconic,
and so forth. These ideas have beautiful connections with one of the most
important topics in classical projective geometry, the classification of
plane curves in projective geometry. In our own time, these ideas
together with problems involving "modular forms" and so forth have given
rise to an important and difficult new subject, "geometric invariant
Klein's work made it much easier to understand various ways of defining
geometries. For example, it was extremely natural for Minkowski to follow
Klein's thinking in his own study of the symmetry groups of quadratic
forms, which later led to a well-known physical application. Perhaps
trying to distance himself (no pun intended) from the awesome achievements
of his colleague Hilbert in algebraic geometry, Minkowski was also one of
the first mathematicians to extensively study "convex geometry" with a
more general symmetry group than the affine group. This led to
fundamental results like the Brunn-Minkowski theorem which are important
in analysis as well as practically important subjects in applied
mathematics, such as Shannon's information theory.
I should give some hint of the surprises in store when one studies how
Klein's ideas play out for specific finite geometries. Assuming you
already know about finite projective planes, you can ask whether it is
possible to add additional structure to create finite analogues of
symplectic or orthogonal geometries. This is indeed the case, and leads
to finite groups with names like PSp(6,2) for the projective symplectic
group of order 1451520. The corresponding geometry gives additional
structure to five dimensional projective space over the finite field F =
GF(2). In particular, in five-dimensional projective geometry over F we
have 63 points, 651 lines, 1395 2-flats, 651 3-flats, and 63 4-flats.
(The symmetry reflects the well known projective duality, which has an
algebraic counterpart I'll mention below.) In five-dimensional symplectic
geometry over F, the 651 lines split into two groups, 315 "isotropic"
lines and 336 "nonisotropic" lines, and similarly for 2-flats and 3-flats.
The symplectic geometry is "more rigid" because the symmetry group
PSp(6,2) has two orbits on lines, not one as for PGL(6,2). That is, in
projective geometry, all lines are equivalent, but in symplectic geometry,
there are two distinct kinds of line.
That's not the surprise. The surprise is that this group PSp(6,2) also
has a primitive action on 28 points, which has an amazing and beautiful
connection with a certain "configuration" of lines studied in algebraic
geometry by none other than Felix Klein, the 28 bitangents of general
quartic curve in complex projective three space CP^3!
(The terms "action", "orbit", "k-transitive" and "primitive" are all
explained in the books by Cameron and by Neumann et al. which were cited
By the way, connections between finite projective geometries and algebraic
coding theory are well known in information theory, but pursuing this line
of thought we find another and far more fundamental connection with
information theory. I'll say a bit more further down.
> What's involved in interpreting an abstract group geometrically?
Well, it depends upon what kind of group you mean and what kind of
invariants you are trying to interpret. Mathematicians like Klein, Lie,
Poincare, Hilbert, and Minkowski were particularly interested in groups
and invariants relevant to certain constructions in differential
equations, algebraic geometry, and what we now call modular forms.
Here is a problem closely resembling problems actually studied by Lie and
Klein around 1872:
In studying the symmetries of an ordinary differential equation, according
to Lie, you should examine a certain Lie algebra, given up to isomorphism
by a presentation in terms of generators and Lie brackets. (The
corresponding Lie group will be the group of "internal symmetries" of the
differential equation; Lie thought of this group as analogous to the
Galois group of a polynomial in one variable.) Now you can try to find a
representation of your Lie algebra as a Lie algebra of vector fields on
R^2 (or some other manifold). Equivalently, you can try to realize the
"abstract group" via a "concrete action" by diffeomorphisms on R^2.
Next, describe the invariant ring of G and interpret the invariants
geometrically in terms of a suitable "Kleinian geometry" on the plane (or
another manifold). Fit this geometry into Klein's hierarchy of
To take the example most likely to be familiar to the average reader,
suppose you have encountered an "abstractly" defined Lie algebra generated
by A,B,C where
[A,B] = 0
[B,C] = -A
[C,A] = -B
(To tell the truth, this is not one of the three dimensional Lie algebras
you would be likely to encounter as the Lie algebra of the point symmetry
group of a second order differential equation, as Lie himself showed; I
could have chosen another example but this one is surely the most
familiar, so let's use it instead.)
Now, these relations can be realized "concretely" by taking A,B,C to the
the following three vector plane fields:
A = @/@x (translation: flowlines horiz. lines)
B = @/@y (translation: flowlines ver. lines)
C = -y @/@x + x @/@y (rot. about origin: flowlines circles)
You may recognize this algebra as e(2), the Lie algebra of E(2), the three
dimensional Lie group of (proper) euclidean isometries. Here, E(2) is the
matrix Lie group consisting of all matrices of form
[ cos(p) -sin(p) | q ]
[ sin(p) cos(p) | r ]
[ 0 0 | 1 ]
It acts on the plane via transformations T(p,q,r) of form
(x,y) -> ( x cos(p) - y sin(p) + q, x sin(p) + y cos(q) + r )
Using elementary reasoning (again due to Lie), we easily find that the
-most- general second order ODE invariant under E(2) is
kappa = --------------
(1 + y'^2)^(3/2)
All such invariants are in fact functions of this one, so this is the
"fundamental invariant", and it now acquires the geometric interpretation
of a "differential path invariant", a quantity giving an invariant
geometric description of a curve with the (noninvariant!) coordinate
description y = y(x). You may recognize this quantity as none other than
the -euclidean path curvature-! (It can be rewritten in parametric form,
which may be more familiar.) And the "curves of constant kappa", say
kappa = 1, are of course the unit circles, so in "euclidean distance
geometry on R^2", there is a notion of "circles of radius r". This
fundamental connection with the differential geometry of curves may
indicate why Klein's conversations in Paris with Darboux played an
important role in the development of his ideas.
I have been discussing "one-point differential invariants" but of course
one can also look for "m-point polynomial invariants", an even more
classical notion. In euclidean geometry we have of course an obvious two
point invariant, and given m, the ring of m point invariants is generated
by the pairwise squared distances. An important point from commutative
algebra is illustrated by the ring of 6 point invariants. This is
generated by the 6 pairwise squared distances, and no smaller set, but
these are not algebraically independent; they satisfy a single cubic
relation which I won't bother to write out.
This relation is an example of a "syzygy", in fact, a Cayley-Menger
syzygy. Syzygies also arise in projective geometry; for example bivectors
on R^4 satisfy a quadratic syzygy, called a Grassmann-Pluecker syzygy.
Indeed, syzygies arise all over the place in invariant theory and
commutative algebra. The term "syzygy" is due to Sylvester, and as the
name implies, they are sexy creatures indeed. One of Hilbert's most
stunning achievements was his famous Syzygy Theorem, one of the
fundamental results in what we would now call commutative algebra.
"Commutative" since in polynomial rings, xy = yx and so forth. But it is
natural to also consider invariants which are exterior forms, which brings
us into another algebraic realm. An important topic in invariant theory
is handling these exterior form invariants, which are important for
Cartan's work, which brings us back to topics like Bianchi groups.
I have now sketched how, starting from a Lie algebra defined abstractly,
we can obtain a Kleinian geometry having the corresponding Lie group as
its symmetry group. You should look up the other Bianchi classes, study
the books by Olver, and repeat this exercise for the other eight classes.
Bianchi type VI_0 is similar but gives E(1,1), the group of isometries of
the Minkowski plane E^(1,1). The second order differential path invariant
is the Minkowski path curvature. Bianchi type VIII gives the Lie groups
isomorphic to SL(2,R) or SO(1,2), the group of isometries of H^2, and
Bianchi IX gives SO(3), the group of isometries of S^2. Bianchi V is the
group Hom(2) of "euclidean homotheties on R^2", and so on. The
relationship between Hom(2) and E(2) illustrates the Cayley-Klein duality
mentioned above. And on and on-- it's a lot of fun.
Of course, this stuff is not limited to three dimensional Lie groups, nor
to actions on two dimensional manifolds.
For example, there is a four dimensional Lie algebra, sim(2), which can be
realized as the Lie algebra of a transformation group on the plane by
throwing in a fourth generator,
D = x @/@x + y @/@y
This gives Sim(2), the symmetry group of "euclidean similarity geometry of
the plane". Now kappa is -no longer invariant- (too rigid), but we have a
new third order differential path invariant,
y''' (1 + y'^2) - 3 y''^2 y'
lambda = ----------------------------
Do you see how this is related to kappa? Consequently, curves with
vanishing "similarity path invariant" are curves of -constant euclidean
path curvature-, i.e circles, so now there is a concept of "circle" and a
concept of "line", but -no concept of "circle of curvature one"-. We can
say that -euclidean distance geometry is "more rigid than" euclidean
similarity geometry-, because -it admits more invariants-.
This was one of Klein's greatest insights: we have a whole partially
ordered hierarchy of geometries, with the increasing "rigidity"
corresponding to inclusion and quotient relations among the releveant
invariant rings of the symmetry groups. I plan to come at this central
theme of Kleinian geometry in a different way below.
I have been postponing talking about homogeneous spaces (see the article
by Millman above), but I really need to say a tiny bit about them here.
Given a finite dimensional Lie group G, each closed subgroup H is
associated with the "Kleinian geometry" on the "homogeneous space" G/H,
which is the space of cosets made into a finite dimensional smooth
manifold. Klein, Lie, Poincare, Hilbert, and Minkowski discovered or were
otherwise familiar with some examples you may have encountered:
E(n)/SO(n) ~ congruence geometry on E^n,
where E(n) ~ R^n |x SO(n),
SO(n+1)/SO(n) ~ congruence geometry on S^n
SO(n+1)/O(n) ~ congruence geometry on RP^n = S^n/antipodes
SO+(1,n)/SO(n) ~ congruence geometry on H^n
(technically, I am restricting to "proper" transformations here)
GL(n,R)/(appropriate block triangular subgroup)
~ Grassmannian of k-flats passing through a point
O(n)/(similar) ~ euclidean geometry on same
SL(2,R)/SL(2,Z) ~ unimodular trellises in R^2
(this homog. space is homeomorphic to the complement of a
trefoil knot; it was studied by Klein and Minkowski as
part of their geometry of continued fractions)
This is only the tip of the iceberg; some of the most interesting examples
come from very classical algebraic geometry. See the textbook by Harris
for an introduction emphasizing the role of symmetry groups.
There are other important examples such as conformal geometry on E^n or
E^(1,n), in which the parent group can be characterized as the point
symmetry group of an important partial differential equation from
mathematical physics. I should probably add that the Poincare group is
the symmetry group of a -system- of partial differential equations, the
Maxwell equations. This group is the semidirect product of translations
on R^4 with the Lorentz group, just as the euclidean group E(3) is the
semidirect product of translations on R^3 with the rotation group SO(3).
The book by Sharpe mentioned above focuses on generalizations of this
pattern to other geometries and other classical groups. This may again
help indicate why the Lorentz and Poincare groups arose first (in several
different ways) in the context of Kleinian geometry, and why it was
natural in the late nineteenth century for people like Poincare to think
of them as symmetry groups of things like quadratic forms or systems of
differential equations, without neccessarily appreciating the kinematical
interpretation of the Poincare group viz the Galillei group. See the
books by Sharpe and Olver for more details.
Let me pause to sketch how the Cartanian geometry unifies Kleinian
geometries--- which are always "homogeneous" geometries--- with Riemannian
geometry, an "inhomogeneous" notion of geometry based upon euclidean
The basic idea of Riemannian geometry is essentially to make a fiber
bundle over an open ball in R^n with fibers isomorphic to SO(n). Then,
choose a smooth section in this fiber bundle to define a connection. Via
Cartan's work on exterior calculus on manifolds, this connection has a
curvature, which turns out to just be another way of expressing the
Riemann curvature tensor. It describes how when a vector is parallel
transported in a small closed loop based at some point in the ball, it
will come back rotated, an effect which varies from basepoint to
basepoint, hence the "inhomogeneity" of this kind of geometry.
The basic idea of Cartanian geometry is to simply replace SO(n) with
another finite dimensional Lie group. We obtain things like Moebius
geometry, an inhomogeneous notion of geometry in which parallel transport
of a vector in a closed loop can bring back it back with a -different
length- as well as pointing in a different direction--- which may sound
familiar to physics students! As the name suggests, this geometry
actually dates right back to Moebius, a contemporary of Gauss!
Mathematicians have an awkward habit of discovering important ideas in an
illogical order-- how ironic.
So far I have not even mentioned one of Klein's most important insights,
which involves a precise connection between the algebraic notion of the
lattice of subgroups (of a symmetry group) and the corresponding geometry.
Actually, there are several ways in which precise correspondences between
certain subgroups and certain geometrical subobjects come up in Kleinian
geometry. One involves -setwise stabilizers-. The most beautiful
expression of this line of thought involves the parabolic subgroups of
reflection groups; see the resources mentioned above.
Did I mention that it is no accident that of the principal creators of
this theory, Cartan, Weyl, and Coxeter, the first two were also intimately
involved with the development of general relativity? To use an example
familiar to readers here, the Lorentz group is a six dimensional Lie group
SO+(1,3) which can be "locally realized" via an action on the plane,
although it is much more natural to extend this to an action on the
Riemann sphere. In the subgroup lattice we find a four dimensional closed
subgroup isomorphic to Sim(2), and thred dimensional subgroups isorphic to
SO(3), SL(2,R), E(2), etc. Here SO+(1,3)/Sim(2) gives euclidean conformal
geometry on S^2 and has the physical significance discovered by Penrose
(an algebraic geometer by training). The appearance of familiar Kleinian
geometries as subquotients in the lattice of closed subgroups has physical
significance which can be obscured by historical differences between
physics and math terminology, like "Wigner's little groups" rather than
"stabilizer" (of a null vector or spacelike vector respectively).
A second fundamental correspondence with information theoretic ideas
involves -pointwise- stabilizers. This also gives that part of Kleinian
geometry which generalizes an (easy but important) part of Galois theory,
which may suggest why Klein's conversations in Paris with Jordan also were
were a key influence in the development of the ideas of both Klein and Lie
For example, going back to Klein's hierarchy, if we draw the lattice of
(point) stabilizer subgroups of PGL(3,R) -up to conjugacy-, we find this
lattice is in "Galois duality" with the lattice of "flats" in RP^2, as
indicated in this sketch
where ** means "two points", L* means "a line and a point not on the
line", etc. If you compare this with the lattice of stablizer subgroups
for Sim(2) and E(2), you can see in detail how the corresponding
geometries are progressively more rigid. For example, in euclidean
congruence geometry, the distinction between ** and L vanishes. Why is
this? Well, if you don't know the identity of an unknown transformation g
in E(2), but you -do- know how g moves any two distinct points on a line
L, then you know how g moves -all- the points on L! But in projective
geometry, if you don't know the identity of an unknown transformation h in
PGL(3,R), but you -do- know how h moves any two distinct points on the
line L, then there still remains "one degree of freedom" in moving other
points on L. This single degree of freedom measures the amount
information you still need to acquire in order to learn exactly how h
moves L: one real variable's worth. In other words, projective geometry
is rather literally less rigid than euclidean geometry.
As this suggests, the classical notion of "degrees of freedom" which
arises in this kind of analysis is simply the -dimension- of the
homogeneous space obtained by quotienting the parent group by the
appropriate point stabilizer subgroup (called a "complexion"). In our
example, the conditional complexion of L given two points on L is a one
dimensional homogeneous space. Because quotient relations among the
complexions obey laws formally analogous to the laws obtained by Shannon
1948 in his axiomatic characterization of his statistical notion of
"entropy", it is inevitable that Kleinian "degrees of freedom" should
behave formally like entropies.
Thus, in Kleinian geometry of matrix groups we have a kind of
algebraicogeometric entropy, which is simply a numerical invariant of an
underlying and more fundamental concept, the complexion. Passing to the
underlying level of the complexions themselves, we can restate our
observations about the increased rigidity of congruence geometry viz.
projective geometry by characterizing "conditional complexions" which are
analogous to Shannon's "conditional entropies". We can also see how
information is naturally acquired in "chunks".
By the way, this way of thinking is not really very far removed from that
of Hilbert, who knew very well that there are various equivalent ways of
specifying a conic:
* name distinct five points on the conic,
* name four distinct points and the tangent at one of them,
* name three distinct points and the tangents at two of them.
It is natural for us to say that these are three ways of specifying
"equivalent amounts of information" in the geometry of CP^2. The
interesting thing is how this is reflected in the corresponding
homological algebra; see Hal Schenk, Computational Algebraic Geometry.
It is instructive to look at the stabilizer lattice for higher dimensional
projective geometry for a nice connection with Young diagrams and
conjugacy classes of symmetric groups (see the stuff discussed by Baez for
the reason why this is not really surprising), and also to look at finite
projective geometries, which have very similar lattices except when the
field is GF(2). In the latter case, we can count the subgroups in each
conjugacy class, which can actually be a convenient way of counting the
number of "configurations" of a given kind.
For example, for PGL(4,3) the pattern is roughly indicated by this diagram
(you should draw in partial order corresponding to Young diagrams in
columns, as suggested by the spacing):
This means that there are 40 points, 130 lines, and 40 2-flats in F^3
where F = GF(3). There are 4680 configurations of a line and a point not
on a line. Each of these configurations is contained in a unique 2-flat,
and each the 40 2-flats contains 117 such configurations. The conditional
complexion obtained by taking the suitable subquotient in this lattice
describes the additional information we must supply, if we are given the
motion of a a line and a point not on the line under an unknown element of
the group PGL(4,3), in order to determine the motion of the 2-flat which
this configuration "spans".
Another example: we can figure out how much information we gain about the
motion of a line L under an unknown projective transformation if are told
the motion of a distinct line L'. And so forth, for whatever combination
of flats should strike your fancy. Here the "indices" are multiplicative
analogues of Shannonian entropies, so their logarithms behave formally
just like Shannonian entropies.
You can easily find permutation groups for which the complexions have an
obvious connection with Boltzmann entropies (logs of multinomial
coefficients), which you can connect with Boltzmann's primitive notion of
statistical mechanics. (As is well known, in a suitable limit, Boltzmann
entropies and conditional entropies reduce to Shannon entropies and
You can also describe in these information theoretic terms the problem of
solving a permutation puzzle such as a Rubik's cube, where the ability to
identify an unknown group element in bite sized chunks of information may
be easier to appreciate. (It is of course the one we need to restore the
cube to its original configuration!)
Let me end with a cautionary remark regarding the history of mathematics.
I hope that the discussion above has persuaded readers that the issues
which were of concern to people like Lie, Klein, Minkowski, Poincare,
Hilbert and Cartan were far, far richer than most people today are likely
to recognize, unless they have read a substantial fraction of the material
To be precise, I hope it is now clear why
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it would be -utterly impossible- to appreciate, much less discuss
intelligently, the mathematical/physical work of Hilbert, Poincare and
their contempories without being familiar with the phenomena in "the
theory of equations" (now distributed over algebra, algebraic geometry,
invariant theory, and parts of analysis) which concerned them.
I suppose this illustrates one of the major difficulties with the history
of mathematics: even professional historians find it difficult to learn
all the late 19th century background, much less the work since then (e.g.
by Elie Cartan and his heirs) which provides the modern context needed to
explain why Klein remains as important in our time as he was in his own.
Sometimes, when I see people who -vastly- overestimate their qualification
to pontificate about a given topic, it is difficult not to enjoy a few
laughs and leave it at that. I hope a whole buncha recent posters on
sci.physics.research (you know who you are, people!) appreciate that I
took the time here for a more elevated response. I did so because I
recognize that some of the responsibility for the inappropriate obscurity
of Kleinian geometry even today rests with Klein himself, who in an ideal
world would have laid a solid foundation for later expositors of his
important, influential, but largely underappreciated and often
"T. Essel" (banner: "killer app" circa 1960?)