"Lecture" for Week 12

Analysis and topology

My comments this week will refer to those topics that are not in the lecture title. I'm sure that Allen and Stillwell, respectively, have told you more than you really wanted to read in one week about analysis and topology.

Groups, continuous and simple

Although the title of Chapter 23 is "Simple Groups", my primary interest in it is in the continuous, or Lie, groups and the related concept of a Lie algebra. These belong more to analysis and geometry than to algebra. I have screened out from the reading assignment the sections on finite simple groups, except for two at the end about the completion of the classification of all finite simple groups, one of the famous achievements of the last third of the 20th century. (Of course, you may need to skip over some of the sentences in those sections that assume an understanding of the earlier skipped sections.) Unfortunately, Stillwell's discussion of Lie algebras starts in the middle of one of the skipped sections (and is rather vague anyway), so I owe you an alternative introduction to Lie algebras. If you have more background in algebra and less in analysis and geometry, you may prefer Stillwell's material to mine; if so, go thither.

More about quaternions, vectors, and rotational Lie algebras

With this and the previous week under our belts, we are ready to return to the double life of quaternions as both three-dimensional vectors (plus an extra component) and rotations acting upon those vectors. The main point, as I've already mentioned more than once, is that the rotation group of Rn is a manifold of dimension n(n-1)/2 -- so its Lie algebra is a vector space of that dimension -- and that when n = 3 these two dimensions are the same. This is what makes 3-dimensional vector analysis (in particular, the cross product and curl operations) work, in a way that has no precise analogue in other dimensions. It also makes quaternions work.

Today we regard J. Willard Gibbs as the founder of modern, practical vector analysis. P. G. Tait, one of Hamilton's most enthusiastic followers, is quoted as writing "Even Prof. Willard Gibbs must be ranked as one of the retarders of quaternion progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster, compounded of the notations of Hamilton and of Grassmann." [Stillwell does not mention Grassmann, so let's not go there.] Well, I'm sorry, from a contemporary point of view it is Hamilton's quaternion formalism that appears to be a hermaphrodite monster, compounded of Grassmann and Gibbs (the vector space) and Lie, Klein, and Pauli (the rotational Lie algebra). [Pauli (p. 426) is the physicist who popularized the Cayley matrices as the fundamental basis for the Lie algebra of SO(3) in quantum mechanics. More accurately, Pauli's matrices differ from Cayley's (apart from the identity matrix) by a factor of i.]

The details of the correspondence between the vector space and the Lie algebra are tied up with the connection between SO(3) and another group, SU(2), comprising the rotations of 2-dimensional complex space. These groups have the same Lie algebra, abstractly, although realized as 3 x 3 and 2 x 2 matrices in the respective cases. SU(2) is twice as big as SO(3). [Technically: SO(3) is not simply connected, so it has a nontrivial universal covering, which turns out to have two sheets and to be isomorphic to SU(2). In the other direction, the mapping from SU(2) onto SO(3) is a homomorphism that is not an isomorphism.] In an angle parametrization of a curve (one-dimensional subgroup) etA in SU(2), t = 2π is not the same point as t = 0, although they map into the same rotation in SO(3). To get back where you started, you need to go on to 4π.

3-dimensional vectors can be identified with certain 2 x 2 matrices by identifying the standard basis vectors i, j, k with the corresponding Cayley matrices (see p. 426 and the exercises in Secs. 22.3 and 22.4). A rotation of the 3-vectors can be implemented by a similarity transformation x -> uxu^{-1}, where u is a matrix in SU(2). Obviously, u and -u give the same result; this is the 2-sheeted covering staring us in the face. When all the algebra and trig of this is worked out, it turns out that the most natural angle to parametrize u is half of the angle of rotation in SO(3) (reflecting identities such as sin(2t) = 2 sin t cos t). According to the article

S. L. Altmann, "Hamilton, Rodrigues, and the quaternion scandal", Math. Mag. 62 (1989) 291-308,

which I strongly recommend, Hamilton had a habit of sweeping this factor of 2 under the rug, and a more accurate understanding of what was going on was reached independently by O. Rodrigues.

Controversy! Controversy!
Altmann's interpretation has been challenged by J. Pujol, who is much more favorable to Hamilton:
  1. "Hamilton, Rodrigues, Gauss, quaternions, and rotations: a historical reassessment", Commun. Math. Anal. 13 (2012), No. 2, 1-14.
  2. "On Hamilton's nearly-forgotten early work on the relation between rotations and quaternions and on the composition of rotations", Amer. Math. Monthly (2014) 515-522.
I have not yet had time to read these articles carefully enough to have an informed opinion on them.