Set-theoretical solutions to the quantum Yang-Baxter equation
نویسندگان
چکیده
منابع مشابه
Set-theoretical Solutions to the Quantum Yang-baxter Equation
In the paper [Dr], V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider “set-theoretical” solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set X × X, where X is a fixed set. In this paper we study such solutions, which in addition satisfy the unitarity and nondegeneracy conditions. We discuss the...
متن کاملOn Set-theoretical Solutions of the Quantum Yang-baxter Equation
In the paper [Dr] V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider “set-theoretical” solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set X ×X , where X is a fixed set. In this note we study such solutions, which satisfy the unitarity and the crossing symmetry conditions – natural conditions ...
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We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang-Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside structure. Moreover, we introduce the notion of a foldable solution, which extends the one of a decomposable solution. AMS Subject Classification: 16T25, 20F36.
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Because of this, a solution of (1) gives rise to a linear representation of the braid group Bn on V⊗n for every n. In [D], Drinfel’d raised the question of finding set-theoretical solutions of the YangBaxter equation. Specifically, we consider a set S and an invertible map R : S×S → S×S. We think of the Yang-Baxter equation (1) as an equality of maps from S×S×S to S×S×S. As in the linear case, ...
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The notion of a geometric crystal was introduced and developed recently in [2], motivated by the needs of representation theory of p-adic groups. It is shown in [1, 2] that some particular geometric crystals give rise to interesting set-theoretical solutions R of the quantum YangBaxter equation, which satisfy the involutivity (or unitarity) condition RR = 1 (more precisely, they give rise to “r...
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ژورنال
عنوان ژورنال: Duke Mathematical Journal
سال: 1999
ISSN: 0012-7094
DOI: 10.1215/s0012-7094-99-10007-x