Dimension leaps

Question

Many mathematical areas have a notion of "dimension", either rigorously or naively, and different dimensions can exhibit wildly different behaviour. Often, the behaviour is similar for "nearby" dimensions, with occasional "dimension leaps" marking the boundary from one type of behaviour to another. Sometimes there is just one dimension that has is markedly different from others. Examples of this behaviour can be good provokers of the "That's so weird, why does that happen?" reaction that can get people hooked on mathematics. I want to know examples of this behaviour.

My instinct would be that as "dimension" increases, there's more room for strange behaviour so I'm more surprised when the opposite happens. But I don't want to limit answers so jumps where things get remarkably more different at a certain point are also perfectly valid.

Answer 1

Say that a square matrix $u$ with entries in a unital $\mathrm{C}^*$-algebra is a magic unitary if its entries are projections which sum to the identity on each row and column: $$\sum_{k=1}^N u_{ik}=\sum_{k=1}^Nu_{kj}=1.$$

Let $C(S_N^+)$ be the universal $\mathrm{C}^*$-algebra generated by the entries of a magic unitary $u\in M_N(C(S_N^+))$.

Then:

For $N\leq 3$, $C(S_N^+)$ is commutative and finite dimensional; but for $N\geq 4$, $C(S_N^+)$ is non-commutative and infinite dimensional.

It is a nice exercise to prove this at $N=3$.

The significance of this is contained in the quip:

There are no quantum permutations on three or fewer symbols.

To explain all this, including the choice of notation, please see the book, Quantum Permutation Groups, by Teo Banica.

An alternative reference is the survey:

T. Banica, J. Bichon, and B. Collins, Quantum permutation groups: a survey, Banach Center Publ. 78 (2007), 13-34.

The original reference is:

S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195–211

Answer 2

The spaces of sequences of real or complex numbers, $(l^p,||·||_p)$, are not pre-Hilbert spaces unless $p=2$.

Answer 3

Isn't the Frobenius theorem on real division algebras an example?

Answer 4

The max number of points interconnected (every-to-every) by lines of any curvature, such that no line crosses any other line. For $\mathbb{R}^2$ it is only 4 points (smth. like Mercedes symbol) - why 4 and not 3 or 5? How many points are possible to connect in such way in $\mathbb{R}^3$? (I suggest, infinite number, but it is interesting to look at a proof). What are some special properties of the Euclidean $\mathbb{R}^3$ such that the number of interconnected points jumps from 4 in $\mathbb{R}^2$ to infinity in $\mathbb{R}^3$?