On the Addition of Quantum Matrices

Details On the Addition of Quantum Matrices On the Addition of Quantum Matrices

1993-08-30

arxiv.org/abs/hep-th/9308148v1

J.Math.Phys. 35 (1994) 2617-2632

Physics

High Energy Physics

High Energy Physics - Theory

24 pages

Scientific paper

10.1063/1.530527

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.

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