The binary cubic generic Clifford algebra has PI degree three, its irreducible representations are one- or three-dimensional, and the Clifford algebra of a cubic form is Azumaya exactly when the form's coefficients avoid the affine twisted cubic.
Azumaya loci and discriminant ideals of PI algebras
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abstract
We prove that, under mild assumptions, for all positive integers $\ell$, the zero set of the discriminant ideal $D_{\ell}(R/Z(R); tr)$ of a prime polynomial identity (PI) algebra $R$ coincides with the zero set of the modified discriminant ideal $MD_{\ell}(R/Z(R); tr)$ of $R$. Furthermore, we prove that, when $\ell$ is the square of the PI-degree of $R$, this zero set is precisely the complement of the Azumaya locus of $R$. This description is used to classify the Azumaya loci of the mutiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of the top discriminant ideal of a prime PI algebra $R$ coincides with the singular locus of the center of $R$, provided that the discriminant ideal has height at least 2, $R$ has finite global dimension and $R$ is a Cohen-Macaulay module over its center.
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A note on generic Clifford algebras of binary cubic forms
The binary cubic generic Clifford algebra has PI degree three, its irreducible representations are one- or three-dimensional, and the Clifford algebra of a cubic form is Azumaya exactly when the form's coefficients avoid the affine twisted cubic.