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- W2000506540 abstract "We study finite set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over $C$ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra $Acal(C,X,r)$ having a $q$-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of finite abelian group $Gcal$ of left actions on $X$. We study the structure of $Acal(C,X,r)$ and show that they have a $bullet$-product form `quantizing' the commutative algebra of polynomials in $|X|$ variables. We obtain the $bullet$-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed $Gcal$-module (over any field $k$). We provide first steps in the noncommutative differential geometry of $Acal(k,X,r)$ arising from these results. As a byproduct of our work we find that every such level 2 solution $(X,r)$ factorises as $r=fcirctaucirc f^{-1}$ where $tau$ is the flip map and $(X,f)$ is another solution coming from $X$ as a crossed $Gcal$-set." @default.
- W2000506540 created "2016-06-24" @default.
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- W2000506540 date "2008-06-18" @default.
- W2000506540 modified "2023-09-27" @default.
- W2000506540 title "Quantum spaces associated to multipermutation solutions of level two" @default.
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