Deformation of certain quadratic algebras and the corresponding quantum semigroups

LetV be a finite-dimensional vector space. Given a decompositionV⊗V=⊕ _i=1,…n I _i , definen quadratic algebrasQ(V, J _(m)) whereJ _(m)=⊕ _i≠m I _i . There is also a quantum semigroupM(V; I ₁, …,I _n ) which acts on all these quadratic algebras. The decomposition determines as well a family of associative subalgebras of End (V ^⊗k ), which we denote byA _k =A _k (I ₁,…,I _n ),k≥2. In the classical case, whenV⊗V decomposes into the symmetric and skewsymmetric tensors,A _k coincides with the image of the representation of the group algebra of the symmetric groupS _k in End(V ^⊗k ). LetI _i,h be deformations of the subspacesI _i . In this paper we give a criteria for flatness of the corresponding deformations of the quadratic algebrasQ(V, J _(m),h ) and the quantum semigroupM(V;I _1,h ,…,I _n,h ). It says that the deformations will be flat if the algebrasA _k (I ₁, …,I _n ) are semisimple and under the deformation their dimension does not change. Usually, the decomposition intoI _i is defined by a given semisimple operatorS onV⊗V, for whichI _i are its eigensubspaces, and the deformationsI _i,h are defined by a deformationS _h ofS. We consider the cases whenS _h is a deformation of Hecke or Birman-Wenzl symmetry, and also the case whenS _h is the Yang-Baxter operator which appears by a representation of the Drinfeld-Jimbo quantum group. Applying the flatness criteria we prove that in all these cases we obtain flat deformations of the quadratic algebras and the corresponding quantum semigroups. Partially supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences.