arXiv subjects

Kurt Trampel

Publications and source records attributed to Kurt Trampel.

3 recordsLinked to original sources

arXiv2023

Poisson geometry and Azumaya loci of cluster algebras

There are two main types of objects in the theory of cluster algebras: the upper cluster algebras ${\boldsymbol{\mathsf U}}$ with their Gekhtman-Shapiro-Vainshtein Poisson brackets and their root of unity quantizations ${\boldsymbol{\mathsf U}}_\varepsilon$. On the Poisson side, we prove that (without any assumptions) the spectrum of every finitely generated upper cluster algebra ${\boldsymbol{\mathsf U}}$ with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and give an explicit description of it. On the quantum side, we describe the fully Azumaya loci of the quantizations ${\boldsymbol{\mathsf U}}_\varepsilon$ under the assumption that ${\boldsymbol{\mathsf A}}_\varepsilon = {\boldsymbol{\mathsf U}}_\varepsilon$ and ${\boldsymbol{\mathsf U}}_\varepsilon$ is a finitely generated algebra. All results allow frozen variables to be either inverted or not.

math.RT↗

arXiv2020

Root of unity quantum cluster algebras and discriminants

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.

math.QA↗

arXiv2018

Noncommutative discriminants via Poisson primes

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.

math.RA↗