TY - JOUR
T1 - Deformation of certain quadratic algebras and the corresponding quantum semigroups
AU - Donin, J.
AU - Shnider, S.
PY - 1998
Y1 - 1998
N2 - Let V be a finite-dimensional vector space. Given a decomposition V ⊗ V = ⊕i=1,...n Ii, define n quadratic algebras Q(V,J(m)) where J(m) = ⊕1≠m Ii. There is also a quantum semigroup M(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 by Ak = Ak(I1,..., In), k ≥ 2. In the classical case, when V ⊗V decomposes into the symmetric and skewsymmetric tensors, Ak coincides with the image of the representation of the group algebra of the symmetric group Sk in End(V⊗k). Let Ii,h be deformations of the subspaces Ii. 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; I1,h,..., In,h). 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 semisimple operator S on V ⊗ V, for which Ii are its eigensubspaces, and the deformations Ii,h 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.
AB - Let V be a finite-dimensional vector space. Given a decomposition V ⊗ V = ⊕i=1,...n Ii, define n quadratic algebras Q(V,J(m)) where J(m) = ⊕1≠m Ii. There is also a quantum semigroup M(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 by Ak = Ak(I1,..., In), k ≥ 2. In the classical case, when V ⊗V decomposes into the symmetric and skewsymmetric tensors, Ak coincides with the image of the representation of the group algebra of the symmetric group Sk in End(V⊗k). Let Ii,h be deformations of the subspaces Ii. 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; I1,h,..., In,h). 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 semisimple operator S on V ⊗ V, for which Ii are its eigensubspaces, and the deformations Ii,h 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.
UR - http://www.scopus.com/inward/record.url?scp=0032440912&partnerID=8YFLogxK
U2 - 10.1007/bf02897067
DO - 10.1007/bf02897067
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AN - SCOPUS:0032440912
SN - 0021-2172
VL - 104
SP - 285
EP - 300
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -