Let $\Bbbk$ be a algebraically closed field of characteristic zero. A graded Artinian local $\Bbbk$-algebra is $(A,\mathfrak{m},\bigoplus A_i)$ such that $(A,\mathfrak{m})$ is an Artianian local $\Bbbk$-algebra and $A=\bigoplus_{i=0}^dA_i$ is a graded ring structure such that $\mathfrak{m}=\bigoplus_{i=1}^dA_i$.
We can fix dimensions $n_i:=\dim A_i$ and then $(A,\mathfrak{m},\bigoplus A_i)$ defines a multiplicative structure on $V:=V_1\oplus\cdots\oplus V_d$ for $V_i:=\Bbbk^{n_i}$. We choose bases $$e_{i,1},\cdots,e_{i,n_i}\in V_i$$ and the multiplication on $V$ determines $(x_{i,\mu,j,\nu}^{\xi})$ via $$e_{i,\mu}\cdot e_{j,\nu}= \sum_{\xi=1}^{n_{i+j}}x_{i,\mu,j,\nu}^{\xi}e_{i+j,\xi} $$ and we impose polynomials describing associativity (and even commutativity). We can choose another bases and under which the multiplication is represented by $(x')_{i,\mu,j,\nu}^{\xi}$. It is easy to see the transformation group of graded basis is $G:=\prod_{i=1}^d\mathrm{GL}(V_i)$.
We have now a GIT problem $G\curvearrowright X$, where $X=\{(x_{i,\mu,j,\nu}^{\xi}):\text{associativity}\}$. We can twist the action by a character $$\chi:\prod_{i=1}^d\mathrm{GL}(V_i)\to\mathbb{G}_m,\quad (g_1,\cdots,g_d)\mapsto \prod_{i=1}^d\det(g_i)^{\theta_d}$$for some $\theta=(\theta_1,\cdots,\theta_d)\in\mathbb{Z}^d$. The twisted GIT quotient is $$X//_\chi G:=\mathrm{Proj}\big(\bigoplus_{m\in\mathbb{N}}\mathcal{O}(X)^{G=\chi^m}\big)$$where$$\mathcal{O}(X)^{G=\chi^m}=\{f\in\mathcal{O}(X):f(g.p)=\chi(g)^mf(p)\text{ for all }(g,p)\in G\times X\}.$$
The notion of GIT (semi)stability therefore induces a $\theta$-(semi)stability for graded Artianian local $\Bbbk$-algebras. We can then call $X//G$ the moduli space of $\theta$-semistable graded Artianian local $\Bbbk$-algebras.
My questions
What are non-GIT descriptions of $\theta$-stability and $\theta$-semistability?
Are there literatures studying this moduli problem?
