Deformations of quadratic algebras and the corresponding quantum semigroups

Let V be a finite dimensional vector space. Given a decomposition V ⊗ V = ⊕i Ii, define n quadratic algebras (V, Jm) where Jm = ⊕i6=mIi. This decomposition defines also the quantum semigroup M(V ; I1, ..., In) which acts on all these quadratic algebras. With the decomposition we associate a family of associative algebras Ak = Ak(I1, ...In), k ≥ 2. In the classical case, when V ⊗ V decomposes into the symmetric and skewsymmetric tensors, Ak coincides with the group algebra of the symmetric group Sk. Let Iih be deformations of the subspaces Ii. In the paper we give a criteria for flatness of the corresponding deformations of the quadratic algebras (V [[h]], Jih and the quantum semigroup M(V [[h]]; I1h, ..., Inh). It says that the deformations will be flat if the algebras Ak(I1, ..., In) are semisimple and under the deformation their dimension does not change. Usually, the decomposition into Ii is defined by a given Yang-Baxter operator S on V ⊗ V , for which Ii are its eigensubspaces, and the deformations Iih are defined by a deformation Sh of S. We consider the cases when Sh is a deformation of Hecke or Birman-Wenzl symmetry, and also the case when Sh 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. 1 Quadratic algebras, quantum semigroup, and notations Let V be a module over a ring A, I a submodule of V ⊗2 = V ⊗ V . Denote by I i,k the submodule in V ⊗n of the type V ⊗ · · · ⊗ I ⊗ · · · ⊗ V where I occupies the positions i, i+ 1. Set I = ∑ i I i,k and I = ∩iI . So, I = I = I. Let V ∗ be the dual module to V . Denote by I the submodule of V ∗ which consists of all linear mappings φ : V → A such that φ(v) = 0 for v ∈ I.