I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus).
Motivation for the question:
I've been introduced to some properties of $\varepsilon_{ijk}$, e.g. antisymmetry (i.e. if you swap two indices, then the $\varepsilon_{...}$ is negated). Also, cyclic permutations of indices are allowed ($\varepsilon_{ijk}=\varepsilon_{kij}=\varepsilon_{jki}$).
These properties seemed familiar: namely, they satisfy two of the conditions for unit quaternion multiplication. When multiplying unit quaternions, $i, j, k$, we may cyclically permute them ($ijk=kij=jki=-1$), and swapping the order in which we multiply unit quaternions negates the result (e.g. $ij=-ji, jk=-kj, ...$).
Question:
Since $\varepsilon_{ijk}$ is used to represent the cross product of two vectors, is there some inherent relationship between the cross product and quaternion multiplication (and if not/so, why?), or is this just a coincidence?
Any links to some already-established result (if it exists) would also be very helpful!
Yes. If we let a vector be written as the coefficients of a quaternion with zero "real" component, i.e. $[a,\ b,\ c] = ai+bj+ck$, then the cross product is simply the quaternion product with the real part omitted. See also: http://en.wikipedia.org/wiki/Cross_product#Quaternions
You may have heard along the way that vector cross products only exist in 3 and 7 dimensions. Why 3 and 7? Because we can "mimic" the cross product in $\mathbb{R}^3$ with quaternions, and likewise we can use octonions to mimic the cross product in $\mathbb{R}^7$.
Why not any other dimension? As it turns out, octonions represent the highest-dimension normed division algebra. So we can go no higher!