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

TL;DR

This paper explores Leavitt path algebras constructed from quivers satisfying quantum Yang-Baxter, Hecke, and RTT relations, linking quantum algebra structures with graph-based algebraic models.

Contribution

It introduces a novel construction of Leavitt path algebras incorporating quantum relations, connecting quantum matrix algebras and quantum spheres through quiver-based models.

Findings

Realization of quantum matrix algebra as subalgebra

Coordinate algebra of quantum spheres as Zhang twist

Intrinsic generation of quantum relations from quiver adjacency matrices

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-Szyma\'{n}ski 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.