## Abstract

Let V be a finite-dimensional vector space. Given a decomposition V ⊗ V = ⊕_{i=1,...n} I_{i}, define n quadratic algebras Q(V,J_{(m)}) where J_{(m)} = ⊕_{1≠m} I_{i}. There is also a quantum semigroup M(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 by A_{k} = A_{k}(I_{1},..., I_{n}), k ≥ 2. In the classical case, when V ⊗V decomposes into the symmetric and skewsymmetric tensors, A_{k} coincides with the image of the representation of the group algebra of the symmetric group S_{k} in End(V^{⊗k}). Let I_{i,h} be deformations of the subspaces I_{i}. In this paper we give a criteria for flatness of the corresponding deformations of the quadratic algebras Q(V, J_{(m),h}) and the quantum semigroup M(V; I_{1,h},..., I_{n,h}). It says that the deformations will be flat if the algebras A_{k}(I_{1},..., I_{n}) are semisimple and under the deformation their dimension does not change. Usually, the decomposition into I_{i} is defined by a given semisimple operator S on V ⊗ V, for which I_{i} are its eigensubspaces, and the deformations I_{i,h} are defined by a deformation S_{h} of S. We consider the cases when S_{h} is a deformation of Hecke or Birman-Wenzl symmetry, and also the case when S_{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.

Original language | English |
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Pages (from-to) | 285-300 |

Number of pages | 16 |

Journal | Israel Journal of Mathematics |

Volume | 104 |

DOIs | |

State | Published - 1998 |