Are there Lie groups in fractional dimension?

Question

Consider the Sierpiński triangle. It has dimension $\log_23$.

So does it have any rotational group associated with it? e.g. a Lie group $SO(\log_2 3)$ ?

Or are there any such things as Lie groups in fractional dimension?

Answer 1

Every Lie group is, by definition, a differentiable manifold. Therefore, its dimension is always a non-negative integer.

Answer 2

We can have groups that have fractional dimension. Consider the cantor set which consists of all numbers between $0$ and $1$ inclusive such that the base representation that real number does not include the digit 1. Examples of valid values in the set include $0.02020202…$ the set can be made into a group by considering the following operation applied digit wise:

$$ 0 + 2 = 2 \\ 2 + 0 = 2 \\ 0 + 0 = 0 \\ 2 + 2 = 0 $$

You end up with a group which has exactly 1 element for every element in the Cantor set. This group isn’t continuous with respect to any reasonable metric because the Cantor set is totally disconnected, but this gives at least a good start. There are higher dimensional cantor like sets (such as the vicsek fractal) which are connected and so the resultant group might at least retain that continuity when assembled.

Also depending on how you intrinsically measure distance between elements in a fractal it’s possible to come up with some gnarly looking group operations that start to resemble Lie groups but aren’t because of the fractal dimension. I will expand on this later when I have more time.