On the addition of quantum matrices

An addition law is introduced for the usual quantum matrices A(R) by means of a coaddition Δ_t=t⊗1+1⊗t. It supplements the usual comultiplication Δt=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, left-invariant vector fields are constructed 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.