On the Addition of Quantum Matrices

TL;DR

This paper introduces a new addition law for quantum matrices using a coaddition operation, forming a braided algebra structure, and explores applications in quantum space vector fields and lattice field theories.

Contribution

It presents a novel coaddition operation for quantum matrices that complements existing structures and leads to braided Lie algebra formations and applications in lattice quantum theories.

Findings

Defined a coaddition law for quantum matrices

Constructed braided Lie algebra of invariant vector fields

Applied to lattice Kac-Moody algebra wave-functions

Abstract

We introduce an addition law for the usual quantum matrices $A(R)$ by means of a coaddition $\underline{\Delta} t=t\otimes 1+1\otimes t$. It supplements the usual comultiplication $\Delta t=t\otimes t$ and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular $m\times n$ quantum matrices. As an application, we construct left-invariant vector fields on $A(R)$ and other quantum spaces. They close in the form of a braided Lie algebra. As another application, the wave-functions in the lattice approximation of Kac-Moody algebras and other lattice fields can be added and functionally differentiated.