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Transposed Poisson structures on the $q$-analog Virasoro-like algebras and $q$-Quantum Torus Lie algebras

Jie Lin, Chengyu Liu, Jingjing Jiang

TL;DR

The paper investigates transposed Poisson structures on two q-deformations: the q-analog Virasoro-like algebra and the q-Quantum Torus Lie algebra, distinguishing generic $q$ from primitive roots of unity. It proves that for generic $q$, the q-Virasoro-like algebra lacks nontrivial $ rac{1}{2}$-derivations and thus has no nontrivial transposed Poisson structure, while the q-Quantum Torus Lie algebra has nontrivial $ rac{1}{2}$-derivations but still no nontrivial transposed Poisson structure. At roots of unity, the q-Virasoro-like algebra acquires nontrivial $ rac{1}{2}$-derivations and a nontrivial transposed Poisson structure on its extended form, whereas the q-Quantum Torus Lie algebra continues to lack any nontrivial transposed Poisson structure despite nontrivial $ rac{1}{2}$-derivations. Overall, the work delineates when transposed Poisson structures exist in these quantum deformations and provides explicit constructions in the root-of-unity case for the q-analog Virasoro-like algebra.

Abstract

We investigate the transposed Poisson structures on both the $q$-analog Virasoro-like algebra and $q$-quantum torus Lie algebra considering the cases where $q$ is generic and where $q$ is a primitive root of unity, respectively. We establish the following results: When $q$ is generic, there are no non-trivial $\frac{1}{2}$-derivations and consequently, no non-trivial transposed Poisson algebra structures exist on the $q$-analog Virasoro-like algebra. Meanwhile, the $q$-quantum torus Lie algebra does possess non-trivial $\frac{1}{2}$-derivations but lacks of a non-trivial transposed Poisson structure. When $q$ is a primitive root of unity, both the $q$-analog Virasoro-like algebra and the $q$-quantum torus Lie algebra possess non-trivial $\frac{1}{2}$-derivations. We present the non-trivial transposed Poisson algebra structure for the $q$-analog Virasoro-like algebra. However, the $q$-quantum torus Lie algebra lacks of a non-trivial transposed Poisson structure.

Transposed Poisson structures on the $q$-analog Virasoro-like algebras and $q$-Quantum Torus Lie algebras

TL;DR

The paper investigates transposed Poisson structures on two q-deformations: the q-analog Virasoro-like algebra and the q-Quantum Torus Lie algebra, distinguishing generic from primitive roots of unity. It proves that for generic , the q-Virasoro-like algebra lacks nontrivial -derivations and thus has no nontrivial transposed Poisson structure, while the q-Quantum Torus Lie algebra has nontrivial -derivations but still no nontrivial transposed Poisson structure. At roots of unity, the q-Virasoro-like algebra acquires nontrivial -derivations and a nontrivial transposed Poisson structure on its extended form, whereas the q-Quantum Torus Lie algebra continues to lack any nontrivial transposed Poisson structure despite nontrivial -derivations. Overall, the work delineates when transposed Poisson structures exist in these quantum deformations and provides explicit constructions in the root-of-unity case for the q-analog Virasoro-like algebra.

Abstract

We investigate the transposed Poisson structures on both the -analog Virasoro-like algebra and -quantum torus Lie algebra considering the cases where is generic and where is a primitive root of unity, respectively. We establish the following results: When is generic, there are no non-trivial -derivations and consequently, no non-trivial transposed Poisson algebra structures exist on the -analog Virasoro-like algebra. Meanwhile, the -quantum torus Lie algebra does possess non-trivial -derivations but lacks of a non-trivial transposed Poisson structure. When is a primitive root of unity, both the -analog Virasoro-like algebra and the -quantum torus Lie algebra possess non-trivial -derivations. We present the non-trivial transposed Poisson algebra structure for the -analog Virasoro-like algebra. However, the -quantum torus Lie algebra lacks of a non-trivial transposed Poisson structure.

Paper Structure

This paper contains 8 sections, 13 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transposed Poisson structures on $q$-Virasoro-like algebras
  4. The case where $q$ is not a root of unity for $q$-Virasoro-like algebra
  5. The case where $q$ is a root of unity for $q$-Virasoro-like algebra
  6. Transposed Poisson structures on $q$-Quantum Torus Lie algebra
  7. The case where $q$ is generic for the $q$-Quantum Torus Lie algebra
  8. The case where $q$ is a root of unity for the $q$-Quantum Torus Lie algebra

Key Result

Lemma 2.1

bib3 Let $\left(L,\left[\cdot,\cdot\right]\right)$ be a Lie algebra and $\cdot$ a new binary (bilinear) operation on L. Then $\left(L,\cdot ,\left[\cdot,\cdot\right]\right)$ is a transposed Poisson algebra if and only if $\cdot$ is commutative and associative and for every $z\in L$ the multiplicatio

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • ...and 14 more