Poisson trace orders

TL;DR

This paper introduces Poisson trace orders, establishing their properties, compatibility with traces, and constructing various examples including quantum groups and algebras, linking trace and Poisson approaches in representation theory.

Contribution

It defines Poisson trace orders, proves their properties, and constructs numerous examples, bridging two independent approaches in the study of algebraic orders.

Findings

All regular and reduced traces are compatible with Poisson order structures.

Discriminant ideals of Poisson trace orders are Poisson ideals.

Zero loci of discriminant ideals are unions of symplectic cores.

Abstract

The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley--Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3 and 4-dimensional Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.