Quantum and Braided Linear Algebra

TL;DR

This paper develops a covariant quantum (braided) linear algebra framework by interpreting quantum matrices via Zamalodchikov algebras, introducing braid statistics, and constructing braided groups with novel conjugation actions.

Contribution

It introduces a detailed interpretation of quantum matrices through Zamalodchikov algebras, incorporating braid statistics to develop a fully covariant braided linear algebra framework.

Findings

Braided tensor products realize braided matrices B(R).

Inner product leads to an invariant quantum trace.

Braided groups act on themselves by conjugation, unlike quantum groups.

Abstract

Quantum matrices $A(R)$ are known for every $R$ matrix obeying the Quantum Yang-Baxter Equations. It is also known that these act on `vectors' given by the corresponding Zamalodchikov algebra. We develop this interpretation in detail, distinguishing between two forms of this algebra, $V(R)$ (vectors) and $V^*(R)$ (covectors). $A(R)\to V(R_{21})\tens V^*(R)$ is an algebra homomorphism (i.e. quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. We show that if $V(R)$ and $V^*(R)$ are endowed with the necessary braid statistics $\Psi$ then their braided tensor-product $V(R)\und\tens V^*(R)$ is a realization of the braided matrices $B(R)$ introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from $B(R)$ act on themselves by conjugation in a way impossible for the quantum groups obtained from $A(R)$.