Including generated 3rd party PHP library in Magento2

EDIT (see updated part (d) below).

Below are some answers to subparts of the whole question.

a). "...What is known about the maximal (or supremal) value of such norms?"

It is known that for the Frobenius norm $\| [A,B] \|_F \le \sqrt{2}\| A \|_F\|B\|_F$, and this bound is tight. More generally, Wenzel and Audenaert (Impressions of convexity --- An illustration for commutator bounds, math.FA-1004.2700v1) prove commutator bounds for Schatten norms. These bounds assume the form

$$ \| AB-BA\|_p \le C_{p,q,r} \|A\|_q\|B\|_r, $$

where $\|X\|_p$ denotes the Schatten-p norm. Moreover, Wenzel and Audenaert establish the tightest possible values for the constant $C_{p,q,r}$.

For lower-bounds, much less is known. See this popular MO question for pointers.

b). "Are there quantifiers of noncommutativity that can also account for higher-order effects?"

Here, you could again repeatedly invoke the above norm based commutator bounds. More interestingly, one could check to see how many "large" terms there are in the corresponding Baker-Campbell-Hausdorrf series (though this sounds like overdo to me)

c). "Maximally noncommutative": Here, one way to characterize maximally noncommutative matrices would be to find examples that make the commutator norm inequality above into an equality. For explicit constructions, please see the cited paper of Wenzel and Audenaert.

d). "other ways of quantifying noncommutativity?" --- this is open ended, but one other way to quantify nc could be to see how many "bits of information" are required to encode the difference $AB-BA$. Such an "information complexity" measure might not be that easy to make precise though.

EDIT: If one restricts the class of operators / matrices of interest, then more interesting measures of noncommutativity are possible, including those with a physical interpretation. The most notable examples come from quantum information theory, where measures of noncommutativity have been of interest for quite long. Here, instead of arbitary complex matrices, a measure called skew entropy has been used to quantify noncommutativity of a positive operator $A$ with respect to a fixed Hermitian operator $K$ (see [1] for details). This is defined as, \begin{equation*} S(A,K) := \frac{1}{2}\mbox{trace}[A^{1/2},K]^2. \end{equation*} The above quantity was introduced by Wigner and Yanase, and was generalized later by Dyson to \begin{equation*} S_t(A,K) := \frac{1}{2}\mbox{trace}[A^t,K][A^{1-t},K],\quad 0 < t < 1. \end{equation*} (The second measure is also interesting for the fact that it's concavity (in $A$ was conjectured by Dyson, and proved as a consequence of the famous Lieb concavity theorem).

References

[1]: Positive definite matrices. R. Bhatia. Princeton University Press. 2007.

Have a look at this paper:

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&ved=0CHEQFjAH&url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi%3D10.1.1.54.7075%26rep%3Drep1%26type%3Dpdf&ei=_1mtT8vHDI-1hAf3kJifDA&usg=AFQjCNG4ku3ERsScCZe9qdfYhT5mYGcaZA

One way to measure commutativity defect,as discussed in the paper is by the rank of $[A,B]$. If $rank([A,B])=1$, $A,B$ are called a Laffey pair.

This approach is based on the fact that commuting matrices have a simultaneous triangularization. See also Commuting Matrices and the Weak Nullstellensatz.

I don't imagine what I will write below has anything to do with free probability, but it outlines an algebraic approach along the lines of your third and fourth paragraph. Namely, I'd suggest looking at work of Kapranov, and Feigin and Shoikhet:

Noncommutative geometry based on commutator expansions, http://arxiv.org/abs/math/9802041

On $[A,A]/[A,[A,A]]$ and on a $W_n$-action on the consecutive commutators of free associative algebra http://arxiv.org/abs/math/0610410

and several follow-up papers featuring the term "lower central series" in either the title or the abstract. The idea is that there are a variety of natural filtrations on any algebra which capture precisely this idea that operators may not commute but may have identities involving higher commutators which are a sort of weakened commutativity.

A neat example of this is due to Feigin-Shoikhet and says that the algebra of even degree differential forms on $\mathbb{C}^n$ (equipped with a certain quantized form) is the universal quotient of the free algebra $A_n$ on $n$ generators such that $[a,[b,c]]=0$ for all $a,b,c \in A_n$ (but $[a,b]\neq 0$ in general). That such a general quotient of the free algebra gives you something so directly related to commutative algebra (and in particular exhibiting polynomial growth in degree) is rather interesting. In particular, the case you asked about is when $n=2$, and you get just the algebra of functions and two forms, with product $f\ast g = fg + df\wedge dg$.

One answer to 4 is that one can define a locally free algebra to be an algebra $A$, such that the completion of $A$ at the lift from $A_{ab}$ of any maximal ideal is free. For instance, the algebra $A_3 / <x^2+y^2+z^2-1>$ is locally free of rank two, even though it is not free, because one can easily show that its completions at

$<(x-x_0),(y-y_0),(z-z_0)>$

is a (completed) free algebra in two generators for any $(x_0,y_0,z_0)$ satisfying the defining equation. These (not just this example, but this class of locally free algebras) behave similarly to free algebras, and it was showed by Etingof that their universal quotient by triple commutators as in the above paragraph is again (a quantization of) even-degree differential forms (but now on the corresponding variety).