Twisted conjugation braidings and link invariants [Elektronische Ressource] / vorgelegt von María Guadalupe Castillo Pérez

Twisted conjugation braidingsand link invariantsDissertationzurErlangung des Doktorgrades (Dr. rer. nat.)derMathematisch-Naturwissenschaftlichen Fakultat¨derRheinischen Friedrich-Wilhelms-Universit¨at Bonnvorgelegt vonMar´ıa Guadalupe Castillo P´erezausMexico CityBonn, Februar 2009AngefertigtmitGenehmigungderMathematisch-NaturwissenschaftlichenFakulta¨t der Rheinischen Friedrich-Wilhelms-Universitat¨ Bonn.1. Referent: Prof. Dr. C.-F. Bod¨ igheimer2. Referent: Prof. Dr. Catharina StroppelPromotionsdatum: 5. Juni 2009Erscheinungsjahr: 2009AbstractThis work is about link invariants arising from enhanced Yang-Baxter opera-tors. For each enhanced Yang-Baxter operatorR = (R,D,λ,β) and any braid−ω(ξ) −n ⊗nBr(n) Turaev defined a link invariant T (ξ) = λ β trace(b (ξ)◦D ),R Rwhere ω : Br(n) →Z is a homomorphism and b is the representation of theRArtin braid group Br(n) arising from the solution of the Yang-Baxter equati-on R. Therefore, we first introduce new solutions of the Yang-Baxter equationϕ ⊗2 ⊗2 ϕ −1B : V → V , B (a⊗b) = abϕ(a) ⊗ϕ(a), for V =K[G], ϕ∈ Aut(G),where G is any group. We call these solutions twisted conjugation braidings.Then we give sufficient and necessary conditions for a map D to decide whe-ϕther the quadruple (B ,D,λ,β) is an EYB-operator. Moreover, we prove thatϕthe twisted conjugation braidings B can be enhanced using character theory.These enhancements are called character enhancements.