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| In mathematics, a '''braided''' vectorspace <math>\;V</math> is a [[vectorspace]] together with an additional structure map <math>\tau\;</math> symbolizing '''interchanging''' of two vector [[tensor product|tensor copies]]:
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| ::<math>\tau:\; V\otimes V\longrightarrow V\otimes V \,</math>
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| such that the [[Yang–Baxter equation]] is fulfilled. Hence drawing [[Penrose graphical notation|tensor diagram]]s with <math>\tau\;</math> an '''overcorssing''' 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]].
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| As first example, every vector space is braided via the trivial braiding (simply flipping). A [[superspace]] has a braiding with negative sign in braiding two '''odd''' vectors. More generally, a '''diagonal braiding''' means that for a <math>\;V</math>-base <math>x_i\;</math> we have
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| ::<math>\tau(x_i\otimes x_j)=q_{ij}(x_j\otimes x_i) \, </math> | |
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| A good source for braided vector spaces entire [[braided monoidal category|braided monoidal categories]] with braidings between any objects <math>\tau_{V,W}\;</math>, most importantly the modules over [[quasitriangular Hopf algebra]]s and [[Yetter–Drinfeld category|Yetter–Drinfeld modules]] over finite groups (such as <math>\mathbb{Z}_2</math> above)
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| If <math>V\;</math> additionally possesses an [[braided Hopf algebra|algebra structure inside the braided category]] ("braided algebra") one has a '''braided commutator''' (e.g. for a [[superspace]] the [[Commutator|anticommutator]]):
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| ::<math>\;[x,y]_\tau:=\mu((x\otimes y)-\tau(x\otimes y))\qquad \mu(x\otimes y):=xy</math>
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| Examples of such braided algebras (and even [[braided Hopf algebra|Hopf algbebras]]) are the [[Nichols algebra]]s, that are by definition generated by a given braided vectorspace. They appear as quantum Borel part of [[quantum group]]s and often (e.g. when finite or over an abelian group) possess an [[Root system|arithmetic root system]], multiple [[Dynkin diagram]]s and a [[Poincare–Birkhoff–Witt theorem|PBW-basis]] made up of braided commutators just like the ones in [[semisimple lie algebra]]s.
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| <ref name=AS02>Andruskiewitsch, Schneider: ''Pointed Hopf algebras'', New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.</ref>
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| <references/>
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| [[Category:Hopf algebras]]
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| [[Category:Quantum groups]]
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| {{algebra-stub}}
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I am Carlos from Altmunster studying Mathematics. I did my schooling, secured 83% and hope to find someone with same interests in Worldbuilding.
Feel free to visit my webpage :: basement renovation ideas (Learn Alot more Here)