On the Addition of Quantum Matrices

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.

24 pages

References in corpus (5)

• Braided Momentum Structure of the q-Poincare Group
• Quantum and Braided Linear Algebra
• Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map
• Infinite Braided Tensor Products and 2-D quantum Gravity
• Glueing operation for r-matrices, quantum groups and link-invariants of Hecke type

Cited by in corpus (8)

• Algebraic {$q$}-Integration and Fourier Theory on Quantum and Braided Spaces
• q-Euclidean space and quantum group wick rotation by twisting
• Introduction to Braided Geometry and $q$-Minkowski Space
• q-Epsilon tensor for quantum and braided spaces
• Lie Algebras and Braided Geometry
• Duality Principle and Braided Geometry
• The quantum orthogonal mystery
• Differential calculus on quantum groups: Constructive procedure