# Operator algebras associated with unitary commutation relations

## TL;DR

This paper introduces a class of nonselfadjoint operator algebras generated by unitary commutation relations, generalizing analytic Toeplitz algebras of rank 2 graphs, and classifies them via an associated unitary matrix.

## Contribution

It defines and classifies new operator algebras based on unitary commutation relations, extending the theory of analytic Toeplitz algebras of rank 2 graphs.

## Key findings

- Algebras are classified up to isometric isomorphism by the unitary matrix u.
- Generalization of analytic Toeplitz algebras of rank 2 graphs.
- Provides a framework for understanding operator algebras with unitary commutation relations.

## Abstract

We define nonselfadjoint operator algebras with generators $L_{e_1},..., L_{e_n}, L_{f_1},...,L_{f_m}$ subject to the unitary commutation relations of the form \[ L_{e_i}L_{f_j} = \sum_{k,l} u_{i,j,k,l} L_{f_l}L_{e_k}\] where $u= (u_{i,j,k,l})$ is an $nm \times nm$ unitary matrix. These algebras, which generalise the analytic Toeplitz algebras of rank 2 graphs with a single vertex, are classified up to isometric isomorphism in terms of the matrix $u$.