Quasi-associativity and flatness criteria for quadratic algebra deformations |
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Authors: | Joseph Donin Steven Shnider |
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Affiliation: | (1) Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel |
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Abstract: | LetR h be the quantumR-matrix corresponding to a Drinfeld-Jimbo quantum groupU h (G). Suppose a finite dimensional representationM h ofU h (G) is given. TheR h induces an operator onM h ⊗2 andS h , its composition with the standard transposition, is the Yang-Baxter operator. It turns out that the spaceM h ⊗2 admits the decompositionM h =⊕ i n J ih whereJ ih are the eigensubspaces ofS h . Consider the quadratic algebras (M h , E h k ) whereE h k =⊕ i≠k J ih . We prove that all (M h ,E h k ) are flat deformations of the quadratic algebras (V 0,E 0 k ). Let End(M h ;J 1h , …,J nh ) be the quantum semigroup corresponding to this decomposition. Our second result is that this gives a flat deformation of the quantum semigroup End(M 0;J 1,0, …,J n,0). Supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities. |
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