Let $V$ be a vector space over a field $k$ and let $n \geq 2$ be an integer. An element $\mathsf{w}$ of the $n^\text{th}$ tensor power of $V$ is called a twisted superpotential if there exists $\sigma \in \operatorname{GL}(V)$ such that $(\text{id}^{\otimes n-1} \otimes \sigma)\theta(\mathsf{w}) = \mathsf{w}$ where $\theta$ cycles the leftmost factor of an $n$-fold elementary tensor to the right end.
Given such a twisted superpotential (or really, any element of $V^{\otimes n}$, but the algebras obtained by twisted superpotentials are nice, as we will state soon) and an integer $m \leq \ell - 2$, one may define the derivation-quotient algebra, which is the algebra $T(V)/\langle \del^{m} V \rangle$, where $\del^\ell(\mathsf{w})$ denotes the span of results of $\ell$ "formal deletions" of the variables from the left of $\mathsf{w}$.
A result of Dubois-Violette shows that if $A$ is an $m$-Koszul AS regular algebra, then there exist a twisted superpotential $\mathsf{w}_A$ such that $A \cong T(V)/\langle \del^{\ell}(\mathsf{w}_A) \rangle$, where $\ell$ is determined by the degree of the relations of $A$, as well as the global dimension of $A$.
We illustrate with an example.
i1 : kk = ZZ/32009
o1 = kk
o1 : QuotientRing
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i2 : R = skewPolynomialRing(kk,(-1)_kk,{a,b,c,d,e})
o2 = R
o2 : FreeAlgebraQuotient
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i3 : A = ambient R
o3 = A
o3 : FreeAlgebra
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i4 : wR = superpotential R
o4 = a*b*c*d*e + a*b*c*e*d + a*b*d*c*e + a*b*d*e*c + a*b*e*c*d + a*b*e*d*c +
------------------------------------------------------------------------
a*c*b*d*e + a*c*b*e*d + a*c*d*b*e + a*c*d*e*b + a*c*e*b*d + a*c*e*d*b +
------------------------------------------------------------------------
a*d*b*c*e + a*d*b*e*c + a*d*c*b*e + a*d*c*e*b + a*d*e*b*c + a*d*e*c*b +
------------------------------------------------------------------------
a*e*b*c*d + a*e*b*d*c + a*e*c*b*d + a*e*c*d*b + a*e*d*b*c + a*e*d*c*b +
------------------------------------------------------------------------
b*a*c*d*e + b*a*c*e*d + b*a*d*c*e + b*a*d*e*c + b*a*e*c*d + b*a*e*d*c +
------------------------------------------------------------------------
b*c*a*d*e + b*c*a*e*d + b*c*d*a*e + b*c*d*e*a + b*c*e*a*d + b*c*e*d*a +
------------------------------------------------------------------------
b*d*a*c*e + b*d*a*e*c + b*d*c*a*e + b*d*c*e*a + b*d*e*a*c + b*d*e*c*a +
------------------------------------------------------------------------
b*e*a*c*d + b*e*a*d*c + b*e*c*a*d + b*e*c*d*a + b*e*d*a*c + b*e*d*c*a +
------------------------------------------------------------------------
c*a*b*d*e + c*a*b*e*d + c*a*d*b*e + c*a*d*e*b + c*a*e*b*d + c*a*e*d*b +
------------------------------------------------------------------------
c*b*a*d*e + c*b*a*e*d + c*b*d*a*e + c*b*d*e*a + c*b*e*a*d + c*b*e*d*a +
------------------------------------------------------------------------
c*d*a*b*e + c*d*a*e*b + c*d*b*a*e + c*d*b*e*a + c*d*e*a*b + c*d*e*b*a +
------------------------------------------------------------------------
c*e*a*b*d + c*e*a*d*b + c*e*b*a*d + c*e*b*d*a + c*e*d*a*b + c*e*d*b*a +
------------------------------------------------------------------------
d*a*b*c*e + d*a*b*e*c + d*a*c*b*e + d*a*c*e*b + d*a*e*b*c + d*a*e*c*b +
------------------------------------------------------------------------
d*b*a*c*e + d*b*a*e*c + d*b*c*a*e + d*b*c*e*a + d*b*e*a*c + d*b*e*c*a +
------------------------------------------------------------------------
d*c*a*b*e + d*c*a*e*b + d*c*b*a*e + d*c*b*e*a + d*c*e*a*b + d*c*e*b*a +
------------------------------------------------------------------------
d*e*a*b*c + d*e*a*c*b + d*e*b*a*c + d*e*b*c*a + d*e*c*a*b + d*e*c*b*a +
------------------------------------------------------------------------
e*a*b*c*d + e*a*b*d*c + e*a*c*b*d + e*a*c*d*b + e*a*d*b*c + e*a*d*c*b +
------------------------------------------------------------------------
e*b*a*c*d + e*b*a*d*c + e*b*c*a*d + e*b*c*d*a + e*b*d*a*c + e*b*d*c*a +
------------------------------------------------------------------------
e*c*a*b*d + e*c*a*d*b + e*c*b*a*d + e*c*b*d*a + e*c*d*a*b + e*c*d*b*a +
------------------------------------------------------------------------
e*d*a*b*c + e*d*a*c*b + e*d*b*a*c + e*d*b*c*a + e*d*c*a*b + e*d*c*b*a
o4 : A
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i5 : I = derivationQuotientIdeal(wR,3)
o5 = ideal (a*b + b*a, a*c + c*a, b*c + c*b, a*d + d*a, b*d + d*b, c*d + d*c,
------------------------------------------------------------------------
a*e + e*a, b*e + e*b, c*e + e*c, d*e + e*d)
o5 : Ideal of A
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i6 : Igb = NCGB(I,10)
o6 = | ab+ba ac+ca bc+cb ad+da bd+db cd+dc ae+ea be+eb ce+ec de+ed |
1 10
o6 : Matrix A <-- A
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i7 : R' = A/I
o7 = R'
o7 : FreeAlgebraQuotient
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i8 : phi = map(R,R',gens R)
o8 = map (R, R', {a, b, c, d, e})
o8 : RingMap R <-- R'
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i9 : ncKernel phi
o9 = ideal 0
o9 : Ideal of R'
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