In mathematics, a braided vector space is a vector space together with an additional structure map symbolizing interchanging of two vector tensor copies:
such that the Yang–Baxter equation is fulfilled. Hence drawing tensor diagrams with an overcrossing the corresponding composed morphism is unchanged when a Reidemeister move is applied to the tensor diagram and thus they present a representation of the braid group.
As first example, every vector space is braided via the trivial braiding (simply flipping)[clarification needed]. A superspace has a braiding with negative sign in braiding two odd vectors. More generally, a diagonal braiding means that for a -base we have
braided, vector, space, mathematics, braided, vector, spacev, displaystyle, vector, space, together, with, additional, structure, displaystyle, symbolizing, interchanging, vector, tensor, copies, displaystyle, otimes, longrightarrow, otimes, such, that, yang, . In mathematics a braided vector spaceV displaystyle V is a vector space together with an additional structure map t displaystyle tau symbolizing interchanging of two vector tensor copies t V V V V displaystyle tau V otimes V longrightarrow V otimes V such that the Yang Baxter equation is fulfilled Hence drawing tensor diagrams with t displaystyle tau an overcrossing the corresponding composed morphism is unchanged when a Reidemeister move is applied to the tensor diagram and thus they present a representation of the braid group As first example every vector space is braided via the trivial braiding simply flipping clarification needed A superspace has a braiding with negative sign in braiding two odd vectors More generally a diagonal braiding means that for a V displaystyle V base xi displaystyle x i we have t xi xj qij xj xi displaystyle tau x i otimes x j q ij x j otimes x i A good source for braided vector spaces entire braided monoidal categories with braidings between any objects tV W displaystyle tau V W most importantly the modules over quasitriangular Hopf algebras and Yetter Drinfeld modules over finite groups such as Z2 displaystyle mathbb Z 2 above If V displaystyle V additionally possesses an algebra structure inside the braided category braided algebra one has a braided commutator e g for a superspace the anticommutator x y t m x y t x y m x y xy displaystyle x y tau mu x otimes y tau x otimes y qquad mu x otimes y xy Examples of such braided algebras and even Hopf algebras are the Nichols algebras that are by definition generated by a given braided vectorspace They appear as quantum Borel part of quantum groups and often e g when finite or over an abelian group possess an arithmetic root system multiple Dynkin diagrams and a PBW basis made up of braided commutators just like the ones in semisimple Lie algebras 1 Andruskiewitsch Schneider Pointed Hopf algebras New directions in Hopf algebras 1 68 Math Sci Res Inst Publ 43 Cambridge Univ Press Cambridge 2002 This algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Braided vector space amp oldid 1176318502, wikipedia, wiki, book, books, library,