$\newcommand\la{\langle}\newcommand\ra{\rangle}\newcommand\laa{\langle\!\langle}\newcommand\raa{\rangle\!\rangle}$ Assume in the following that every polynomials is a sum of monomials of degree at least two.

For $K$ an algebraically closed(this is probably needed?) field and $f_i$ polynomials, the algebra $A=K[x_i]/(f_i)$ has finite vector space dimension iff the $f_i$ have finitely many solutions and then the vector space dimension of $A$ is equal to the number of solutions.

In particular, because of this result algebras of the form $K[x_i]/(f)$ for a single polynomial $f$ should never be finite dimensional when we have at least two variables.

Let $K\la x_i\ra$ be the non-commutative polynomial ring in $n$ variables $x_i$ and define $K\laa x_i\raa:=K\la x_i\ra/ \la x_i x_j+x_j x_i \ra$ for $i \neq j$ to be the quasi-polynomial ring (or is there already a (better) name for this in the literature?).

So instead of the commutativity relations $x_i x_j=x_j x_i$ we have $x_i x_j =- x_j x_i$.

Interestingly we can have polynomials $f$ such that $K\laa x_i\raa/(f)$ is finite dimensional. Here is an example class of such polynomials. Let $f_{n,a_i}:=\sum\limits_{r=1}^{n-1}{x_r^{a_r}}-x_n^{a_n}$ where $a_i \geq 2$. So for $n=3$ those polynomials are the Fermat polynomials of the form $x_1^{a_1}+x_2^{a_2}-x_3^{a_3}$. Let $A_{n,a_i}:=K\laa x_i\raa/(f_{n,a_i})$.

Question 1: What is the $K$-dimension of $A_{n,a_i}$ depending on the parameters $a_i$ and $n$? When is this algebra finite dimensional?

For $n=3$ the algebra is for example finite dimensional for $a_1=3,a_2=3,a_3=2$ with dimension 26 and infinite dimensional for $a_1=2,a_2=2,a_3=3$. Does the vector space dimension have a number theoretic meaning in case it is finite?

Question 2: For which polynomials $f$ is a quotient of the quasi-polynomial ring $K\laa x_i\raa/(f)$ finite dimensional and what is the $K$-dimension?

Is there a nice interpretation in terms of properties or even certain zeros of $f$?

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