Quantum groups at intersection between algebra and geometry
Staic, Mihai D.
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If ( H, T ) is a pure braided Hopf Algebra and V 1 , V 2 , ..., V n are H modules I construct a representation of the pure braid group P n on V 1 ⊗ V 2 ⊗ ... ⊗ V n . I use this representation to obtain invariants for long knots. Then I show that (anti-)Yeter-Drinfeld modules are the degree 0 (respectively degree 1) component in a braided Z -crossed category. A third result concerns the input information that is needed to construct invariants for 3-manifolds. I give a generalization of the fundamental group and show that in this context is more natural to talk about ternary operation than the usual binary product. This discussion leads to a new cohomology theory which I call the symmetric cohomology.