Deformation of certain quadratic algebras and the corresponding quantum semigroups

Abstract

LetV be a finite-dimensional vector space. Given a decompositionV⊗V=⊕i=1,…nIi, definen quadratic algebrasQ(V, J(m)) whereJ(m)=⊕i≠mIi. There is also a quantum semigroupM(V; I1, …,In) which acts on all these quadratic algebras. The decomposition determines as well a family of associative subalgebras of End (V⊗k), which we denote byAk=Ak(I1,…,In),k≥2. In the classical case, whenV⊗V decomposes into the symmetric and skewsymmetric tensors,Ak coincides with the image of the representation of the group algebra of the symmetric groupSk in End(V⊗k). LetIi,h be deformations of the subspacesIi. 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;I1,h,…,In,h). It says that the deformations will be flat if the algebrasAk(I1, …,In) are semisimple and under the deformation their dimension does not change.