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I'm interested in Sklyanin Algebras or Artin-Shelter regular algebras of type A. These are generated in degree 1 by three variables x,y,z, and have three defining relations in degree 2, which you can get by multiplying the matrix $$M=\begin{pmatrix} cx & bz & ay \\ az & cy & bx \\ by & ax & cz\end{pmatrix}$$ either on the left or right by the row/column vector of variables. (For every triple of numbers $a$, $b$, $c$, excluding a finite set, you get one of these algebras.)

Such algebras $A$ contain a central regular element of degree $3$, let me call it $g$. It has the property that $A/gA$ is the twisted homogeneous coordinate ring of the scheme of point modules of $A$, which is the cubic curve defined by $\det M$. $$g=c(c^3-b^3)y^3+b(c^3-a^3)yxz+a(b^3-c^3)xyz+c(a^3-c^3)x^3$$

This central element $g$ seems somewhat mysterious. In the original paper by Artin-Shelter it was apparently found by a computer search, in the foundational paper by Artin-Tate-Van den Bergh, only its existence (and its properties) are proved.

My question is: is there a conceptual way to get this element $g$? Sorry, it's a little vague. Thanks for any insight into this matter.

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If translation by the point $\tau$ of the elliptic curve $E$ has infinite order, then the center of $A$ is $k[g]$. That's the only intrinsic description of $g$ I know. There might be more known about the two central elements in the Sklyanin algebra of GKdim 4. Paul Smith has some notes that might be useful there: – Graham Leuschke Apr 25 '13 at 20:22

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