Azumaya loci and discriminant ideals of PI algebras

We prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket skein algebras contains the smooth locus, proving a strong form of the Unicity Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as it arises in each setting. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of Frobenius Poisson orders. We use these tools to construct canonical central subalgebras of quantum algebras, and explicitly compute the resulting Azumaya loci we encounter, using a natural nondegeneracy assumption.

pdf/the-quantum-frobenius-for-character-varieties-and-1jd628go65.pdf

The quantum Frobenius for character varieties and multiplicative quiver varieties

On the discriminant of twisted tensor products

Abstract We study a Morita-equivalent version of the Zariski cancellation problem.

A Morita Cancellation Problem

A general criterion is given for when the center of a Taft algebra smash product is the fixed ring. This is applied to the study of the noncommutative discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully determine the inner faithful actions of Taft algebras on quantum planes and quantum Weyl algebras. We compute the discriminant of the corresponding smash product and apply it to compute the Azumaya locus and restricted automorphism group.

Discriminants of Taft Algebra Smash Products and Applications

Actions of cocommutative Hopf algebras

We describe a connection between the subjects of cluster algebras, polynomial identity algebras and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we prove an explicit formula for the discriminant of the integral form over ${\mathbb{Z}}[\varepsilon]$ of each quantum unipotent cells of De Concini, Kac and Procesi for arbitrary symmetrizable Kac-Moody algebras, where $\varepsilon$ is a root of unity.

Root of unity quantum cluster algebras and discriminants

If S is a given regular d-simplex of edge length a in the d-dimensional Euclidean space $$\mathcal {E}$$
 , then the distances $$t_1$$
 , $$\ldots $$
 , $$t_{d+1}$$
 of an arbitrary point in $$\mathcal {E}$$
 to the vertices of S are related by the elegant relation $$\begin{aligned} (d+1)\left( a^4+t_1^4+\cdots +t_{d+1}^4\right) =\left( a^2+t_1^2+\cdots +t_{d+1}^2\right) ^2. \end{aligned}$$
 The purpose of this paper is to prove that this is essentially the only relation that exists among $$t_1,\ldots ,t_{d+1}.$$
 The proof uses tools from analysis, algebra, and geometry.

Distances from the Vertices of a Regular Simplex

We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by specialization from families, such as quantum algebras at roots of unity. It is illustrated with the specializations of the algebras of quantum matrices at roots of unity and more generally all quantum Schubert cell algebras.

Noncommutative discriminants via Poisson primes

If S is a given regular n-simplex, $$n \ge 2$$
 , of edge length a, then the distances $$a_1, \ldots , a_{n+1}$$
 of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation $$\phi _{n+1} (a,a_1,\ldots ,a_{n+1})=0$$
 , where $$\begin{aligned} \phi = \phi _t (x, x_1,\ldots ,x_{n+1}) = \left( x^2+x_1^2+\cdots +x_{n+1}^2\right) ^2 - t\left( x^4+x_1^4+\cdots +x_{n+1}^4\right) . \end{aligned}$$
 The natural question whether this is essentially the only relation was recently and positively answered by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha. The authors made use of the irreducibility of the polynomial $$\phi $$
 in the case when $$n \ge 2$$
 , $$t=n+1$$
 , $$x= a \ne 0$$
 , and $$k = \mathbb {R}$$
 , but supplied no proof, promising to do so in another paper that is turning out to be this one. It is thus the main aim of this paper to establish that irreducibility. In fact, we treat the irreducibility of $$\phi $$
 without restrictions on t, x, a, and k. As a by-product, we obtain new proofs of results pertaining to the irreducibility of the general Cayley–Menger determinant that are more general than those established by C. D’Andrea and M. Sombra.

Irreducibility of the Cayley–Menger determinant and of a class of related polynomials

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