Multipartite $d-$level GHZ bases associated with generalized braid matrices

We investigate the generalized braid relation ($d-$level $N-$body braid relation) and its application to quantum entanglement. By means of finite-dimensional representations of Heisenberg-Weyl algebra, a set of $d^{N}\times d^{N}$ unitary matrix representations satisfying the generalized braid relation can be constructed. Such generalized braid matrices can entangle $d-$level $N-$partite quantum states. Acting the generalized braid matrices on the standard basis, one can obtain a set of maximally entangled basis. Further study shows that such entangled basis can be viewed as the $d-$level $N-$partite Greenberger-Horne-Zeilinger (GHZ) basis.