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Quantum spheres as Leavitt path algebras: Quivers with Quantum Yang-Baxter equation, and Hecke condition

Cody Gilbert, Ashish K. Srivastava

TL;DR

The paper develops a framework to realize quantum spaces inside Leavitt path algebras by embedding QYBE, Hecke, and RTT relations into quivers. It proves a structural classification for QYBE-satisfying quivers via $A^2=\mu A$ and shows how adjacency data gives rise to Hecke $R$-matrices, including a subalgebra isomorphic to $\mathcal{O}_q(M_n)$. It then constructs Leavitt path algebras with RTT relations and uses Zhang twists to realize the odd-dimensional Vaksman–Soibelman sphere $\mathcal{O}(S_q^{2n+1})$ as a twisted corner of such algebras. The results bridge Leavitt path algebras, quantum groups, and noncommutative geometry, providing an algebraic analogue of the Hong–Szymański correspondence.

Abstract

In this paper we study Leavitt path algebras over quivers with relations such as quantum Yang-Baxter equation, Hecke condition, and RTT conditions. This construction allows us to produce examples of Leavitt path algebras that contain quantum matrix algebra as subalgebra and obtain an algebraic analogue of the Hong-Szymański result. In particular, we show that the coordinate algebra over odd-dimensional Vaksman-Soibelman quantum sphere can be realized as the Zhang twist of a Leavitt path algebra over a quiver with such relations. Furthermore, we show that the quantum Yang-Baxter equation, and Hecke condition for our RTT construction can be generated intrinsically from the adjacency matrix of certain quivers.

Quantum spheres as Leavitt path algebras: Quivers with Quantum Yang-Baxter equation, and Hecke condition

TL;DR

The paper develops a framework to realize quantum spaces inside Leavitt path algebras by embedding QYBE, Hecke, and RTT relations into quivers. It proves a structural classification for QYBE-satisfying quivers via and shows how adjacency data gives rise to Hecke -matrices, including a subalgebra isomorphic to . It then constructs Leavitt path algebras with RTT relations and uses Zhang twists to realize the odd-dimensional Vaksman–Soibelman sphere as a twisted corner of such algebras. The results bridge Leavitt path algebras, quantum groups, and noncommutative geometry, providing an algebraic analogue of the Hong–Szymański correspondence.

Abstract

In this paper we study Leavitt path algebras over quivers with relations such as quantum Yang-Baxter equation, Hecke condition, and RTT conditions. This construction allows us to produce examples of Leavitt path algebras that contain quantum matrix algebra as subalgebra and obtain an algebraic analogue of the Hong-Szymański result. In particular, we show that the coordinate algebra over odd-dimensional Vaksman-Soibelman quantum sphere can be realized as the Zhang twist of a Leavitt path algebra over a quiver with such relations. Furthermore, we show that the quantum Yang-Baxter equation, and Hecke condition for our RTT construction can be generated intrinsically from the adjacency matrix of certain quivers.

Paper Structure

This paper contains 5 sections, 13 theorems, 141 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Quivers Satisfying the QYBE
  3. Construction of Hecke $R$-matrices
  4. Application: Leavitt path algebra over quiver with relations
  5. Vaksman-Soibelman Quantum Sphere as Leavitt Path Algebra over quivers with relations

Key Result

Theorem 2.3

Let $A$ be a $n\times n$ matrix, such that $A^3\neq 0$, then if and only if $A^2\otimes A=A\otimes A^2$ if and only if $A^2=\mu A$.

Figures (6)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • ...and 25 more