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Quasi-associativity and flatness criteria for quadratic algebra deformations

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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 ⊗2 h andS h , its composition with the standard transposition, is the Yang-Baxter operator. It turns out that the spaceM ⊗2 h admits the decompositionM h =⊕ n i J ih whereJ ih are the eigensubspaces ofS h . Consider the quadratic algebras (M h , E k h ) whereE k h =⊕ i≠k J ih . We prove that all (M h ,E k h ) are flat deformations of the quadratic algebras (V 0,E k0 ). 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).

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Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, 52900, Ramat Gan, Israel

    Joseph Donin & Steven Shnider

Authors
  1. Joseph Donin
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  2. Steven Shnider
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Correspondence to Joseph Donin.

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Supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences and Humanities.

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Donin, J., Shnider, S. Quasi-associativity and flatness criteria for quadratic algebra deformations. Isr. J. Math. 97, 125–149 (1997). https://doi.org/10.1007/BF02774031

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  • DOI: https://doi.org/10.1007/BF02774031

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