Would it be simpler, pedagogically speaking, if textbook writers introduced root systems as an example of a quandle?

Question

I could never, for the life of me, recall the definition of a root system in Lie theory. It probably doesn't help that I've never taken a course on Lie Theory - the algebra, or the groups, or the differential geometry - even though I was at one of the best universities at the country.

The definition of a root system, I find - if not others, is easily forgettable. However, recently I discovered that root systems were an example of a quandle, the axioms of which go back to Mituhisa Takasaki in 1942, and simply axiomatise the properties of conjugation in groups. I found this useful nugget in the book, Quandles, An Introduction to the Algebra of Knots by Mohammed Elhamdadi and Sam Nelson.

Would it then, not be useful, to include this in a discussion of Lie algebras and their classification?

Answer 1

A root system can be considered as an example of a rack (which is a bit more general than a quandle). Namely, one defines the Coxeter rack $C$ as the Euclidean space $V$ together with an operation $u*v = s_v(u)$, the reflection of $u$ with respect to $v^\perp$. Then a root system (as a subset of $V$) can be considered as a finite subrack of $C$.

Some sources claim that root systems can be defined precisely as the finite subracks of the Coxeter rack. However, if $\Phi$ is a root system with roots of two distict lengths, say, short of length $x$ and long of length $y\neq x$, then scaling all long roots by some arbitrary constant $\lambda$ produces another finite subrack, which is no longer a root system. There are two possible ways to deal with this:

1. Consider these finite subrack of $C$ up to isomorphisms and forget about the particular embedding. This way you can no longer distinguish between the root system and its dual, namely, $B_n$ and $C_n$ type root systems become isomorphic as racks.
2. Equip the embedding $\Phi\subset V$ with some additional structure which allows to reconstruct the lengths ratio. But none of the standard extensions (such as augmented quandles) seem to be natural for this goal. It is possible, I would guess, but certainly not easier than the standard definition of a root system.

One can define a quandle similar to the Coxeter rack $C$ as $V$ together with the operation $u*v=-s_v(u)$, but the same considerations apply.

So while a root system can be considered as an example of a rack or a quandle, this languages does not, apparently, provide an easier way to define them.

Answer 2

You might find useful certain aspects of Alissa Crans's thesis,

• Alissa Crans, Lie 2-Algebras, Dissertation, U. Cal. Riverside, 2004 (link).

For instances, Lie groups and their associated Lie algebras are described as internal unital spindles, which are closely related to quandles.