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Register a new userA note on generic Clifford algebras of binary cubic forms
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We study the representation theoretic results of the binary cubic generic Clifford algebra $mathcal C$, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that $mathcal C$ is a PI algebra of PI degree three and compute its point variety and discriminant ideals. As a consequence, we give a necessary and sufficient condition on a binary cubic form $f$ for the associated Clifford algebra $mathcal C_f$ to be an Azumaya algebra.
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In this paper we present an approach to quadratic structures in derived algebraic geometry. We define derived n-shifted quadratic complexes, over derived affine stacks and over general derived stacks, and give several examples of those. We define the associated notion of derived Clifford algebra, in all these contexts, and compare it with its classical version, when they both apply. Finally, we prove three main existence results for derived shifted quadratic forms over derived stacks, define a derived version of the Grothendieck-Witt group of a derived stack, and compare it to the classical one.
We provide a formula for commputing the discriminant of skew Calabi-Yau algebra over a central Calabi-Yau algebra. This method is applied to study the Jacobian and discriminant for reflection Hopf algebras.
Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. Consider $G$-graded simple algebras $A$ which are finite dimensional and $e$-central over $F$, i.e. $Z(A)_{e} := Z(A)cap A_{e} = F$. For any such algebra we construct a textit{generic} $G$-graded algebra $mathcal{U}$ which is textit{Azumaya} in the following sense. $(1)$ textit{$($Correspondence of ideals$)$}: There is one to one correspondence between the $G$-graded ideals of $mathcal{U}$ and the ideals of the ring $R$, the $e$-center of $mathcal{U}$. $(2)$ textit{Artin-Procesi condition}: $mathcal{U}$ satisfies the $G$-graded identities of $A$ and no nonzero $G$-graded homomorphic image of $mathcal{U}$ satisfies properly more identities. $(3)$ textit{Generic}: If $B$ is a $G$-graded algebra over a field then it is a specialization of $mathcal{U}$ along an ideal $mathfrak{a} in spec(Z(mathcal{U})_{e})$ if and only if it is a $G$-graded form of $A$ over its $e$-center. We apply this to characterize finite dimensional $G$-graded simple algebras over $F$ that admit a $G$-graded division algebra form over their $e$-center.
Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $mathbb Z_2$-graded algebra $E(theta)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $zin E^!_2$. It turns out that the bounded derived category $D^b(text{gr}_{mathbb Z_2}E(theta))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(theta)$ is a semisimple $mathbb Z_2$-graded algebra. The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to the Kn{o}rrer Periodicity Theorem for quadric hypersurfaces. As an application, we recover Kn{o}rrer Periodicity Theorem without using of matrix factorizations.
We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.
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