ANNOUNCEMENTS
MTH 675 — Winter 2023

3/14/23

The final installment of my handwritten lecture notes (not the in-class notes) are available here.

Today is both Pi Day and Einstein's birthday!

3/13/23

I have reorganized and expanded Chapter 8 of GELG on the exceptional Lie groups.

Most details are still (only) in GO, but you may find the new summary in GELG to be helpful.

3/12/23

Two mathematicians are talking on the telephone. Both are in the continental United States. One is in a West Coast state, the other is in an East Coast state. They suddenly realize that the correct local time in both locations is the same! How is this possible?

Give up? Some hints can be found here.

3/5/23

There are several ways one can check whether matrices are linearly independent, such as those being used to generate a Clifford algebra.

• First of all, it is obvious that the set of elementary matrices that each differ from the zero matrix by a single "1" is linearly independent.
• For matrices with a block structure, it's enough to show that matrices with the same nonzero components are independent in each case.
• If the matrices can be represented using tensor products, you can use the fact that $A\otimes B$ can only equal $C\otimes D$ if $A=C$ and $B=D$ (up to scale). So tensor products of independent matrices are themselves independent.
• The Killing form $\tr(AB)$ provides a nondegenerate inner product that can be used to check linear independence (over the reals).

3/4/23

A standard reference on the octonions is the article by John Baez, which is available here.

Here are some additional resources you may find interesting:

• A fascinating and reasonably accessible paper on the applications of $G_2$ to rolling balls can be found here.
• A readable presentation of the classification of simple Lie algebras can be found in Fulton and Harris or Erdmann and Wildon; see the book list.

A Mathematica package for multiplying octonions is available here.

The minimal documentation fails to mention that the octonionic units can be entered as one,i,j,k,kl,jl,il,l.

This version of the package is old, but still appears to work; if you have difficulty, please let me know.

3/3/23

My summary of the Cayley–Dickson process in class today was incomplete, as it is also necessary to specify the products between $\KK$ and $\KK e$.

A paragraph to this effect has been added to the end of the section on the Cayley–Dickson process in Chapter 2 of GELG.

As stated in class, the sedenions, obtained as the next step in the Cayley–Dickson process after the octonions, do indeed have zero divisors. However, they are not a composition algebra, since the inner product is positive definite. Thus, no null elements are available to "rescue" the composition property for zero divisors (as happens, for instance, in $\CC'$).

Further discussion of both the split division algebras and the sedenions can be found in Chapter 5 of GO.

3/2/23

A new section has been added to the appendix of GELG that discusses direct sums and tensor products.

There have also been minor edits to other sections.

2/23/23

The root diagrams for $\mathfrak{b}_3$ and $\mathfrak{c}_3$ in GELG (and their labeling!) have been updated.

Yes, there were missing lines. Please let me know if you spot any further errors.

2/22/23

The online text GELG has been updated.

The examples shown in class today can be found in new sections at the end of Chapter 5.

2/20/23

Some further information about Clifford algebras (and division algebras) can be found in these two papers:

Paper 1,

Paper 2.

2/17/23

The next installment of my handwritten lecture notes (not the in-class notes) are available here.

These notes cover the material through today's class, including the details we omitted.

2/16/23

The discussion of the properties of roots in GELG has been updated based on the discussion during and after class yesterday.

The paper I referred to in class yesterday that constructs some higher-dimensional root diagrams can be found here

The online version uses obsolete technology, so you may prefer the PDF version, which is also available here.

2/10/23

Solutions to the exercise done in class today (except for the case $\theta=\frac\pi6$) can be found on the schedule page.

There is also a new section in GELG that discusses reflection symmetry.

2/7/23

As pointed out on the HW page, the suggested exercises for this week are needed in order to complete the assignment due next week.

We will spend much of class on Friday discussing these exercises.

If you are not able to attend class that day, you are expected to include solutions to these exercises along with your homework..

2/4/23

There is an important subtlety about yesterday's derivation of the connection and curvature:

The formulas $\nabla_XY=\frac12[X,Y]$ and $R(X,Y)Z=-\frac14\bigl[[X,Y],Z\bigr]$ were derived assuming that $X$, $Y$, $Z$ are left invariant. They can be extended to arbitrary vector fields by expanding them with respect to a basis of left-invariant vector fields.

However, the Lie bracket of vector fields is linear over scalars, but NOT over functions.

An elementary example would be the vector fields $x\,\Hat{y}$ and $y\,\Hat{x}$, which do not commute even though $[\Hat{x},\Hat{y}]=0$.

Thus, if the left-invariant basis is $\{e_a\}$, then \[ \nabla_XY = \nabla_{X^a e_a}(Y^b e_b) = X^a\nabla_{e_a}(Y^b e_b) = X^ae_a(Y^b)e_b + X^aY^b\nabla_{e_a}(e_b) = X(Y^b)e_b + \frac12 X^aY^b [e_a,e_b] \] but the last term is not $\frac12[X,Y]$ unless $X$ and $Y$ are left invariant, in which case $X^a$ and $Y^b$ are constant.

The scalar curvature is independent of the basis used to compute it, so the derivation in class suffices to show that it is constant.

2/3/23

The torsion-free condition was stated incorrectly in class today...

The correct condition is $\nabla_XY-\nabla_YX=[X,Y]$, which does not involve functions at all.
This condition is exactly what is needed for the derivation of the connection given in class.

The online notes have been corrected.

As promised, my handwritten lecture notes so far (not the in-class notes) are now available here.

Please do not redistribute these informal notes.

1/27/23

Here's the essence of the rushed construction at the end of today's lecture:

Consider a representation $\rho:\su(2)\longmapsto\mathfrak{gl}(V)$ with natural basis \[ L_z=\frac12\rho(\sigma_z), \quad L_\pm=\frac12\rho(\sigma_\pm)=\frac12\rho(\sigma_x\pm i\sigma_y) \] and commutation relations \[ [L_z,L_\pm]=\pm L_\pm, \quad [L_+,L_-]=2L_z . \] Then if $w\in V$ is an eigenvector of $L_z$, so $L_zw=\lambda w$, then \[ L_z(L_\pm w) = [L_z,L_\pm]w + L_\pm L_z w = \pm L_\pm w + \lambda L_\pm w = (\lambda\pm1)L_\pm w \] so $L_\pm$ raises/lowers the eigenvalue by $1$. But if $V$ has finite dimension, there must be a largest eigenvalue $\lambda$, in which case $L_+ w$ must be $0$. If $V$ is irreducible, ALL eigenvectors of $L_z$ must be of the form $(L_-)^k w$. But there must also be a smallest eigenvalue, corresponding to some $k-1$, so $(L_-)^k w$ must be $0$ for that value of $k$. A straightforward induction argument (see GELG) then shows that \[ L_+(L_-)^k w = \bigl(2k\lambda-k(k-1)\bigr) (L_-)^{k-1} w \] which in turn forces $2k\lambda-k(k-1)=0$, so $\lambda=\frac{k-1}2\in\frac12\mathbb{Z}$. Furthermore, the smallest eigenvalue is $\lambda-2\lambda=-\lambda$.

Any (finite, irreducible) representation of $\su(2)$ can thus be described by a weight diagram consisting of the points from $-\lambda$ to $\lambda$, separated by $1$, with $2\lambda\in\mathbb{Z}$; this representation is referred to as having "spin $\lambda$". This diagram fully captures the algebraic description of $\su(2)$ acting on $V$.

1/20/23

As discussed at the beginning of class today, here are some comments and suggestions regarding assignments.
See also the Ground Rules.

• Assignments will be graded for both content (out of 10) and presentation (out of 5).
• Collaboration is fine (and encouraged), so long as it is acknowledged. Writeups are of course expected to be your own.
• Lengthy and/or obvious arguments can be summarized briefly in words, without providing all details.
• Use complete, grammatically correct sentences in most cases.
• Avoid flowery langauge with little content, but do explain your reasoning.
  "This means that..." doesn't help the reader as much as, say, "This symmetry of the dot product means that...".
• The standard to aim for is whether you could understand what you wrote 5 years from now without other resources.
• It's good practice to include your name, and perhaps the assignment ("HW1"), in the filename – and in the assignment.

1/17/23

You can find out more about the split complex numbers in §5.4 of GO.

You are encouraged to work out the explicit formula for the general element of $\SO(3)$ in terms of Euler angles.
Is there a simple formula for the product of two such elements?

Further discussion can be found on Wikipedia.

1/13/23

Strange but true: The 13th of the month is more likely to be a Friday than any other day of the week!

Give up? Further information is available here.

1/9/23

Please send a short email message to me at tevian@math.oregonstate.edu by Friday 1/13/23.

Please include the following information:

• Your class (senior, 1st-year grad, etc.) and major(s) (math, physics, etc.).
• What is the most advanced algebra class you have taken?
  What is the most advanced geometry class you have taken?
  Feel free to include comments about these classes and/or your preparation in algebra and geometry.
• Why you are taking this class.

I will assume that email is a reliable way to reach you unless you tell me otherwise.

11/20/22

Students can obtain credit for this course under three separate course numbers:

• MTH 675 is the standard course number, intended for graduate students who have either taken MTH 674 or otherwise been exposed to the basic concepts of differential geometry.
• MTH 679 is a topics course, intended for graduate students who have already received credit for (a past incarnation of) MTH 675.
• MTH 405 is a reading course for undergraduates.

The only difference between these alternatives is that some assignments will be optional for students registered for MTH 405.

Deep prior knowledge of differential geometry is not expected. It should be sufficient to be comfortable with changes of variables on parameterized surfaces. More important is comfort with some topics in abstract algebra, such as abstract changes of basis and the use of dual spaces. All of these topics will be (briefly!) reviewed in class before being used.