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Degenerate twisted traces on quantized Kleinian singularities of type A

Zev Friedman, Ben Webster

Abstract

We study the space of non-degenerate traces on quantized Kleinian singularities of type A by studying their complement, the degenerate traces. In particular, we find the dimension of the space of twisted traces as a function of the corresponding automorphism and the quantization parameters, encoded in a polynomial $P$.

Degenerate twisted traces on quantized Kleinian singularities of type A

Abstract

We study the space of non-degenerate traces on quantized Kleinian singularities of type A by studying their complement, the degenerate traces. In particular, we find the dimension of the space of twisted traces as a function of the corresponding automorphism and the quantization parameters, encoded in a polynomial .
Paper Structure (6 sections, 17 theorems, 51 equations)

This paper contains 6 sections, 17 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Degenerate traces
  4. Strong nondegeneracy
  5. Finite-dimensional modules
  6. Lerch transcendents

Key Result

Theorem A

The space of degenerate traces is a subspace whose dimension is $0\leq \delta(P)\leq d-1$ as defined in eq:delta-def, with every integer in this range being realized for every $t$. In particular, we have: In particular, the algebra $\mathcal{A}_P$ has a nondegenerate trace if $t\neq 1$. On the other hand, when $t=1$, a nondegenerate trace exists if and only if $\delta(P)<d-1$, that is, $P$ has a

Theorems & Definitions (29)

  • Theorem A
  • Definition 1
  • Lemma 2: etingofTwistedTraces2021
  • Definition 3
  • Lemma 4: etingofTwistedTraces2021
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • ...and 19 more