Tuesday, January 27th, 2015 | Author: Konrad Voelkel
I was asked to give a talk about division algebras (purpose and classification). This is a rough overview of it, mostly learned from the book of Springer and Veldkamp "Octonions, Jordan Algebras and Exceptional Groups" and the book of Lam "Noncommutative Rings". The book "Numbers" by Ebbinghaus et al. was also nice to read along for references.
We will look mostly at finite dimensional (not neccesarily associative) division algebras over the real numbers.
Instead of these rather loose notes, you may prefer to look at my texed notes in this pdf file.
There are several classes of algebras over a field which we will look at. By "R-algebra" we mean a not necessarily commutative, not necessarily associative, not necessarily unital ring which at the same time is a module over the ring R, so that the ring multiplication is R-linear. We focus on fields R=k, so the word "module" becomes "vector space".
A division algebra over a field is a -algebra which allows division, i.e. for any you have unique elements and , unless . The notation means that is an element such that .
An associative algebra over a field is a -algebra in which the multiplication is associative (what else?). An associative division algebra can be defined to be an associative algebra with a unit element distinct from such that every element has a unique inverse (right- as well as left-inverse).
An example for an associative algebra which is not a division algebra: the split complex numbers with multiplication and consequently . This algebra has zero divisors: .
If the field is finite, the finite dimensional division algebras over it are obviously also finite. In the other direction, if a division algebra over a field is finite, Wedderburn's little theorem shows that and must be finite fields.
The Cayley-Dickson construction is a process to take an associative division algebra and produce a new algebra of twice the dimension. It actually works like the construction that you use to produce the complex numbers from the real numbers: define a multiplication on the direct sum which is again a division algebra. Starting from , the Cayley-Dickson construction produces . Starting from , we get the quaternions and from that the octonions . The next step is no longer a division algebra (the sedenions, of dimension 16).
If we look at the field of real numbers , Frobenius' theorem states that the only finite dimensional associative division algebras over , up to isomorphism, are .
Over any algebraically closed field such as , there are no finite dimensional associative division algebras except itself.
There are infinite dimensional associative algebras: take a Banach space and the linear endomorphisms under composition , then that is a unital associative -algebra (with complete norm, thus a Banach algebra). The only Banach division algebras are , by a theorem of I.Gelfand and S.Mazur.
There are infinite dimensional associative divison algebras, but they don't admit a norm (otherwise they would embed into their completion, which by the previous theorem is finite dimensional). An example was found by Hilbert: take formal Laurent series in one variable over the field and define a new product by extending the rule to all Laurent series. The result is not a field, but an associative division algebra over .
A central simple algebra over a field is a finite dimensional associative algebra which is simple and has as center only . Every simple algebra is a central simple algebra over its center. One should probably say something about Brauer groups here, but leaving the name "Brauer" here should suffice to point anyone in the right direction.
A composition algebra is a unital algebra equiped with a nondegenerate quadratic form (often called the norm) that allows composition, i.e. .
One can show that composition algebras can only occur in dimensions , by an analogue of the Cayley-Dickson procedure, which takes a subalgebra and returns a subalgebra of twice the dimension. One can show that a proper subalgebra is always associative; The doubling procedure, started from the one-dimensional subalgebra generated by the neutral element, gives at some point a non-associative 8-dimensional subalgebra, which must be the whole thing (if the doubling didn't end earlier on).
Over the real numbers, not only appear, but also things like the split complex numbers, split quaternions, etc. In a composition algebra, an element has an inverse if and only if the norm is nonzero.
Now if one wants to classify the finite dimensional division algebras over without the extra structure of a norm or quadratic form or whatever - you have to do algebraic topology. The solution comes in the form of Adams' solution of the Hopf Invariant One problem, which shows that only the classical Hopf fibrations for have Hopf invariant one, and therefore only the euclidean norm-1-spheres in these -vector spaces admit a multiplication (even up to homotopy). Since any division algebra induces such a multiplication, we know there can't be any other. If you want, a reason behind this is seen in Atiyah's proof, which exploits the 8-fold Bott periodicity, which in turn comes from a mod-8 periodicity in the classification of Clifford algebras (so in fact a very algebraic reason).
The reason I'm interested in brushing up on division algebras? Exceptional groups. In the classification of semisimple Lie algebras (or compact Lie groups, or reductive linear algebraic groups), the exceptional cases (other than the "classical" groups) are rather hard to come by. Unless you have the quaternions and octonions. The magic is best seen in Freudenthal-Tits magic squares, but that can be read over at Wikipedia. Enjoy your algebras.