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Abstract 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 obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity. 

Abstract Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category$${\mathcal {C}}$$and$${\mathcal {C}}$$-module category$${\mathcal {M}}$$, we introduce a version of the Drinfeld center$${\mathcal {Z}}({\mathcal {C}})$$of$${\mathcal {C}}$$adapted for$${\mathcal {M}}$$; we refer to this category as thereflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$of$${\mathcal {M}}$$. Just like$${\mathcal {Z}}({\mathcal {C}})$$is a canonical braided monoidal category attached to$${\mathcal {C}}$$, we show that$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$is a canonical braided module category attached to$${\mathcal {M}}$$; its properties are investigated in detail. Our second goal pertains to when$${\mathcal {C}}$$is the category of modules over a quasitriangular Hopf algebraH, and$${\mathcal {M}}$$is the category of modules over anH-comodule algebraA. We show that the reflective center$${\mathcal {E}}_{\mathcal {C}}({\mathcal {M}})$$here is equivalent to a category of modules over an explicit algebra, denoted by$$R_H(A)$$, which we call thereflective algebraofA. This result is akin to$${\mathcal {Z}}({\mathcal {C}})$$being represented by the Drinfeld double$${\operatorname {Drin}}(H)$$ofH. We also study the properties of reflective algebras. Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangularH-comodule algebras, and we examine their corresponding quantumK-matrices; this yields solutions to the qRE. We also establish that the reflective algebra$$R_H(\mathbb {k})$$is an initial object in the category of quasitriangularH-comodule algebras, where$$\mathbb {k}$$is the ground field. The case whenHis the Drinfeld double of a finite group is illustrated. 

Abstract Beginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every pointWof Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s. 

Discriminant ideals of noncommutative algebras $A$, which are module finite over a central subalgebra $C$, are key invariants that carry important information about $A$, such as the sum of the squares of the dimensions of its irreducible modules with a given central character. There has been substantial research on the computation of discriminants, but very little is known about the computation of discriminant ideals. In this paper we carry out a detailed investigation of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in the sense of De Concini, Reshetikhin, Rosso and Procesi, whose identity fiber algebras are basic. The lowest discriminant ideals are the most complicated ones, because they capture the most degenerate behaviour of the fibers in the exact opposite spectrum of the picture from the Azumaya locus. We provide a description of the zero sets of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in terms of maximally stable modules of Hopf algebras, irreducible modules that are stable under tensoring with the maximal possible number of irreducible modules with trivial central character. In important situations, this is shown to be governed by the actions of the winding automorphism groups. The results are illustrated with applications to the group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity. 

We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley–Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley–Hamilton algebra; its fully Azumaya locus is shown to contain a copy of the underlying cluster A \mathcal {A} -variety. Both results are proved in the wider generality of intersections of mixed quantum tori over subcollections of seeds. Furthermore, we prove that all monomial subalgebras of root of unity quantum tori are Cayley–Hamilton algebras and classify those ones that are maximal orders. Arbitrary intersections of those over subsets of seeds are also proved to be Cayley–Hamilton algebras. Previous approaches to constructing maximal orders relied on filtration and homological methods. We use new methods based on cluster algebras. 

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.