Category Archives: knotes

knotes, Notes

Notes on Partitions of Sets

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In the course of another project, I realized that I need a better understanding of partitions of sets. In particular, I was curious why one doesn’t speak of ‘morphisms’ or ‘isomorphisms’ of these. I figured that there was some simple explanation, and that these notions were either untenable or trivial.

As it happens, partitions of sets turned out to be of limited utility in the particular project that had inspired their exploration. However, the subject itself turned out to be richer than I had imagined. Partly for this reason and partly from a desire to be comprehensive and meticulous (though I won’t claim to have succeeded on either account), what started as a few pages of notes quickly burgeoned. The final result is 80-ish pages long. However, I now feel that I have gained a clearer understanding of the subject. Perhaps someone else (or my future self) will find these notes useful, so I’ve decided to post them online.

These notes were written in pieces and revamped several times, so it is quite possible that there is some redundancy in the proofs or that earlier propositions aren’t used to simplify later proofs. As far as I can tell, there is nothing problematic (like circular reasoning) other than some unnecessary verbiage.

I’m sure that there are typos, and no doubt some of the explanations can be tightened or improved. Hopefully, any actual errors which have crept in aren’t too egregious. If you come across typos, confusing language, or downright errors, I would greatly appreciate it if you direct my attention to them. Any other suggestions for improvements are welcome as well.

I’ve included both color and B&W versions of the notes.

Partitions of Sets (PDF, Color)

Partitions of Sets (PDF, B&W)

knotes, Notes, Uncategorized

The Assumption of Equal apriori Probabilities

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As a student, more years ago than I care to count, I was bothered by something we were taught in statistical mechanics. Actually, I was bothered by a lot of what we were taught in stat mech — but I’m going to focus on one particular issue in this post.

If you’re a physics student, you almost certainly have encountered something called the “assumption of equal apriori probabilities”. This says that every microstate in the microcanonical ensemble has an equal probability.

Putting aside whether such an assumption is philosophically or physically justified, we can ask whether the statement itself has any meaning at all. I.e. can we define a natural notion of uniform probabilities for the set of microstates?

It turns out that the answer is quite complicated. In these notes, I’ll focus on the case of a classical system with a finite number of degrees of freedom. In getting to the answer (spoiler: it’s “yes” with a lot of caveats), we’ll get to explore various aspects of measure and probability theory, symplectic manifolds, and the Riesz representation theorem.

These notes are meant to be fairly self-contained. They assume some familiarity with basic topology, manifolds, and measure theory, but anything beyond that is included in the discussion.

I’m sure there are typos and possibly even omissions or mistakes. If you spot one, please let me know. I’d like to improve these notes in any way I can. They are primarily for my future self, blessed as I am with an incredibly poor memory. However, I hope that they may of interest and use to others as well.

I’ve included two versions: one with color-coded note-boxes, and one without. The former is useful for visually distinguishing side-notes from core concepts, and the latter is optimized for black and white printing and use with ereaders.

Assumption of Equal apriori Probabilities — Color version (PDF)

Assumption of Equal apriori Probabilities — B&W version (PDF)

knotes, Notes

Semidirect Products, Group Extensions, Split Exact Sequences, and all that

UPDATED SUBSTANTIALLY 7/6/2023
[Original Version Posted 1/22/2019]

PDF Notes: Semidirect Products, Group Extensions, Split Exact Sequences, and all that

If you suffer physics-brain and don’t know anything about semidirect products or group extensions, but get the sense direct products just aren’t cutting it — these are the notes for you.

Update: On reexamination, my original post on the subject was too brief and simplistic. Among other things, it neglected to explicitly construct the multiplication on a group extension. In the process of doing so, I identified numerous other shortcomings and omissions. As a result, I’ve replaced it with a far more thorough discussion. Unfortunately, at 50 pages it has expanded beyond the point where the primitive wordpress support for latex suffices. The notes now are provided as a pdf (linked at the top and bottom of this post). This has the added benefit of allowing me to the color-code proofs and comments and examples, allowing easier reading of the key elements.

The purpose of this post is to cure one aspect of physics-brain. In a typical physics education, we learn a sloppy version of linear algebra, basic group theory, and (certain) differential equations many times — and little else math-related. As a result, we develop numerous bad habits and suffer from a complete ignorance of certain important areas of mathematics. If two groups are being combined, it’s via a direct product. After all, what else is there? Unfortunately, this only works until it doesn’t — which is pretty much anywhere it matters. For example, even O(3,1) is not a direct product of the 2 copies of Z2 and SO+(3,1). You’ve probably heard of O(3,1). It matters.

Many of the things we naively assume would be direct products actually are semidirect products or general group extensions, and the result can be a lot of confusion when things don’t work as expected. Besides the construction of basic physical groups such as the Poincare group, this also arises in the study of quantum mechanics. Part of the reason we can pretend to work in Hilbert space when the actual state space is a projective Hilbert space is that the projective representations of a group lift to unitary representations of a different group. The latter is a particular group extension (in fact, a central extension) of the original group.

Ignorance of semidirect products and group extensions is quite understandable in a physicist, but also easily cured. These notes are an attempt to do so based on my experience trying to cure myself. They are self-contained, and the only prerequisite is a little group theory. Though I allude to topology in a couple of places, no knowledge of it is necessary.

These notes cover the following:

  • Normal subgroups and quotient groups, the isomorphism thms, various other group-related concepts, and inner/outer automorphisms.
  • Exact sequences, short-exact-sequences (SES’s), splitting of SES’s, central extensions, and the isomorphism classes of SES’s (as well as how they interact with splitting).
  • Introduction to the hierarchy of direct products, semidirect products, and group extensions. We take 3 views of each: (1) an external view in which we build a new group from two distinct groups, (2) an internal view in which we consider the relationship between an existing group and its subgroups, and (3) an SES-view in which we frame things in terms of short exact sequences. Because the external-view is conceptually the most challenging, we give a preview of its regimen.
  • The direct product in all 3 views.
  • Semidirect products in all 3 views.
  • General group extensions in all 3 views.
  • A brief comparison of direct products, semidirect products, and group extensions in all 3 views.
  • An addendum in which we explicitly construct the multiplication on a group extension in gory detail.

A few caveats. There are lots of detailed proofs in the notes. There also probably are typos and possibly errors. I hope to correct and clarify as needed, so please report these to me if you encounter any. The notes were written in passes and phases. There is a lot of intentional replication of effort in the proofs (where it serves the purpose of clarity), but also probably some unintentional replication (for example, if I consolidated some results in a proposition in one place, but fail to use those results in another). Nonetheless, I believe they should be quite informative and are on balance both correct and quite comprehensible. In particular, I tried to avoid “simplifying” things in subsequent passes as my understanding improved (though I certainly did correct things as needed). The danger in doing so is that I would lose sight of the things which confused me when first encountering the concepts — and one purpose of these notes is to address such sources of confusion. After enough passes and simplifications, the entire piece would end up a one page terse Bourbaki-like statement, along with a vague hint at the possibility of an idea of a proof-outline — which anyone worth their salt is expected to easily be able to expand into a 40-page proof. In math, the best way to learn a subject is to already know it. In physics, we take a gentler approach.

There are a couple of key omission — neither mission-central — which I hope to address in future posts. These concern (1) the relationship between semidirect products/group extensions and fiber bundles/principal bundles and (2) a detailed discussion of group cohomology and the classification of group extensions.

PDF Notes: Semidirect Products, Group Extensions, Split Exact Sequences, and all that

knotes, Notes

The (quasi)-Duality of the Lie Derivative and Exterior Derivative

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Lecture1     Lecture2    Lecture3    Lecture4    Lecture5

This is a short set of notes that covers a couple of aspects of duality in differential geometry and algebraic topology. It grew out of an enigmatic comment I encountered, to the effect that the Lie and exterior derivatives were almost-dual in some sense. I wanted to ferret out what this meant, which turned out to be more involved than anticipated. Along the way, I decided to explore something else I never had properly understood: the nature of integration from a topological perspective. This led to an exploration of the equivalence of de Rham and singular cohomology. The notes are in the form of five sets of slides. Originally, they comprised four presentations I gave in a math study group. On tidying, the last set grew unwieldy, so I broke it into two.
  • Lecture1: Review of DG and AT. Types of derivatives on {M}, de Rham Complex, review of some diff geom, Lie deriv and bracket, chain complexes, chain maps, homology, cochain complexes, cohomology, tie in to cat theory.
  • Lecture2: The integral as a map, Stokes’ thm, de Rham’s thm, more about Lie derivs.
  • Lecture3: Recap of de Rham cohomology, review of relevant algebra, graded algebras, tensor algebra, exterior algebra, derivations, uniqueness results for derivations, the interior product.
  • Lecture4: Cartan’s formula, tensor vs direct product, element-free def of LA, Lie coalgebras
  • Lecture5: Quick recap, relation between struct constants of LA and LCA, the choice of ground ring or field, duality of Lie deriv and exterior deriv.
These notes grew organically, so the order of presentation may seem a bit … unplanned. The emphases and digressions reflect issues I encountered, and may be peculiar to my own learning process and the many gaps in my physicist-trained math background. Others may not share the same points of confusion, or require the same background explanations. They were designed for my own use at some future point when I’ve completely forgotten the material and need a bespoke refresher. I.e., a week from now. Although I’ve tried to polish the notes to stand on their own, there are some allusions to earlier material studied in the group. In particular, certain abbreviations are used. Here is a (hopefully) complete list:
  • DG: Differential Geometry
  • AT: Algebraic Topology
  • DR: de Rham
  • {P}: Used for a Principal bundle. Not really used here, but mentioned in passing.
  • PB: Principal Bundle. Not really used here, but mentioned in passing.
  • AB: Associated Bundle. Not really used here, but mentioned in passing.
  • LG: Lie Group. Mentioned in passing.
  • LA: Lie Algebra
  • LCA: Lie Coalgebra (defined here).
  • v.f. Vector fields
  • v.s. Vector space
The 1st 2 lectures focus on the equivalence of de Rham and singular cohomologies via a duality embodied in the integral map, and enforced by Stokes’ and de Rham’s thms. The last 3 lectures focus on the quasi-duality between the Lie derivative and exterior derivative. By quasi-duality we don’t mean to downplay its legitimacy. I didn’t go through all sorts of contortions to call a square a circle just because it sounds elegant. There is a true duality, and a beautiful one. But saying that it is directly between the Lie and exterior derivs is slightly misleading. These notes were constructed over a period of time, and focus on the specific topic of interest. They are by no means comprehensive. Although edited to correct earlier misconceptions based on later understanding (as well as errors pointed out by the math group), the order of development has not been changed. They were written by someone learning the subject matter as he learned it. They may have some mistakes, there may be some repetition of points, and they are not designed from the ground up with a clear vision. Nonetheless, they may prove helpful in clarifying certain points or as a springboard for further study. These notes explain the following:
  • {\int} as a map from the de Rham complex to the singular cochain complex
  • Stokes’ thm as a relationship between de Rham cohomology and singular cohomology
  • The various types of derivations/anti-derivations encountered in differential geometry
  • A review of graded algebras, tensor algebras, exterior algebras, derivations, and anti-derivations.
  • A review of Lie Derivatives, as well as Cartan’s formula
  • A discussion of what the duality of {{\mathcal{L}}} and {d} means
  • A discussion of the two views one can take of {T(M)} and {\Lambda(M)}: as {\infty}-dimensional vector spaces over {\mathbb{R}} or as finite-basis modules over the smooth fns on M. The former is useful for abstract formulation while the latter is what we calculate with in DG. The transition between the two can be a source of confusion.
  • A discussion of why derivations and anti-derivations are the analogues of linearity when we move from one view to the other.
The notes draw from many sources, including Bott & Tu, Kobyashi & Nomizu, and various discussions on stackexchange. A list of references is included at the end of the last set of slides.