Quasi-associativity and flatness criteria for quadratic algebra deformations

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.