We prove the existence of a basis of Poincaré-Birkhoff-Witt type for braided Hopf algebras R generated by a braided subspace V ⊂ P (R) if the braiding on V fulfils a triangularity condition. We apply our result to pointed Hopf algebras with abelian coradical and to Nichols algebras of low dimensional simple Uq(sl2)-modules.

We give a short overview of the recent research perspectives and mathematical foundations in topological quantum computation theory. In particular, we will be interested in braid representation theory, topological invariants of braids, approximation with braiding generators and quantum hashing with the icosahedral group. Mathematics Subject Classification: 16T25; 20F36; 54H13; 57M25; 57M27; 81P...

In a previous work [AS2] we showed how to attach to a pointed Hopf algebra A with coradical kΓ, a braided strictly graded Hopf algebra R in the category Γ Γ YD of Yetter-Drinfeld modules over Γ. In this paper, we consider a further invariant of A, namely the subalgebra R of R generated by the space V of primitive elements. Algebras of this kind are known since the pioneering work of Nichols. It...

To realize the braiding operations of Majorana zero mode in vortex cores a topological superconductor (TSC), novel approach is proposed this letter to replace common tip (or tip-like) method. Instead on top TSC thin film, arrays electrically controllable pining centers are built beneath hence detecting can proceed along with braiding. It does not only increase rate, but also enables be performe...