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In mathematics, a Hopf algebra, H, is quas … In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of such that
* for all , where is the coproduct on H, and the linear map is given by ,
* ,
* , where , , and , where , , and , are algebra morphisms determined by R is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang–Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, ; moreover , , and . One may further show that theantipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element: where (cf. Ribbon Hopf algebras). It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction. If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding . modules over H is braided with braiding .
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In mathematics, a Hopf algebra, H, is quas … In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of such that
* for all , where is the coproduct on H, and the linear map is given by ,
* ,
* , where , , and , where , , and , are algebra morphisms determined by R is called the R-matrix. It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction. If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding . modules over H is braided with braiding .
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