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This should probably be a comment, since it is too basic an observation. Suppose that the elements $f,g\in k_q[x,y]$ are both normal in $k_q[x,y]$ (this, at least a priori is more restrictive than you …

This complex (in simultaneously a more general setting, where you have several elements $t_1,\ldots,t_p$, and a more special setting, because only the case of $R$ being a free algebra was studied then …

I recall your question on a related topic... I am sure that associativity is assumed here (and just omitted because the audience is unlikely to think of any other algebras); as for unitality, won't …

This algebra $R$ is a quadratic Koszul algebra (in fact, it is easily seen to be PBW, that is has a quadratic Groebner basis), from which one can immediately construct a bimodule resolution; the bimod …

Another possibility is to consider group elements rather than commutators. If you take the matrices $A=\mathrm{diag}(1,\xi,\ldots,\xi^{n-1})$ and $B=\begin{pmatrix}0&1&0&\cdots&0&0\\ 0&0&1&\cdots&0&0\ …

There are lots of papers dealing with representation-theoretic questions and universal enveloping algebras using Gröbner bases. Some examples are given by these: 1, 2, 3, 4.

This variety of algebras is well studied. In particular, the description of the underlying vector space of the free algebra in this variety follows immediately from "A note on the T-ideal generated by …

I have just thought of another interesting example, which is a bit peculiar, in that the 3-commutativity property arises for a natural subspace of an algebra which itself does not satisfy any identity …

The problem is that $U_J\frak{su}_2$ as you define it, that is $\bigoplus\limits_{j=0}^{2J}V_{2j+1}$, is NOT the space of all differential operators of order at most $J$. For the latter, Formula (2) i …

Let me assume that the characteristic of the ground field is different from two. Let me start by replacing your identity by something where the existing symmetries are a bit more apparent. I claim tha …

One very algebraic way to approach it (alas not that easily approached computationally) - somewhat related to David Jordan's answer - would be to look at the Lie subalgebra generated by $A$ and $B$ in …