On the Addition of Quantum Matrices

We introduce an addition law for the usual quantummatrices A(R) by means of a coaddition ∆t = t⊗ 1 + 1⊗ t. It supplements the usual comultiplication ∆t = t⊗ 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 × n quantum matrices. As an application, we construct leftinvariant 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.