Deformation of certain quadratic algebras and the corresponding quantum semigroups |
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Authors: | J Donin S Shnider |
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Institution: | (1) Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel |
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Abstract: | 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. |
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Keywords: | |
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