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Quantum groups of GL(2) representation type

Colin Mrozinski

2014
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Journal of Noncommutative Geometry
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We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories. Theorem 1.2. Assume that char(k) = 0. The Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL 2
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... c to that of GL 2 (k) are exactly the G(A, B) with A, B ∈ GL n (k) (n ≥ 2) satisfying B t A t BA = λI n for some λ ∈ k * and such that any solution of the equation X 2 − √ λ −1 tr(AB t )X + 1 = 0 is generic. A particular case of the theorem was already known if one requires the fundamental comodule of H to be of dimension 2 ([Ohn00]). A similar classification (without dimension constraint) was obtained by Bichon ([Bic03] ) in the SL (2) case (the compact SU (2) case had been done by Banica [Ban96]). The SL(3) case with dimension constraints has been studied by Ohn ([Ohn99]). Other related results have been given in the SU (N ) and SL(N ) case by Banica ([Ban98]) and Phung Ho Hai ([Hai00]), in terms of Hecke symmetries. It is worth to note that in principle Theorem 1.2 could be deduced by the combination of Phung Ho Hai's work [Hai00] and Gurevich's classification of Hecke symmetries of rank two [Gur90] . We believe that the present approach, using directly pairs of invertible matrices, is more explicit and simpler. We also give a version of Theorem 1.2 in the compact case. Finally the following theorem will complete the classification of GL(2)-deformations. Theorem 1.3. Assume that char(k) = 0. Let A, B ∈ GL n (k) and let C, D ∈ GL m (k) such that B t A t BA = λ 1 I n and D t C t DC = λ 2 I m for λ 1 , λ 2 ∈ k * . The Hopf algebras G(A, B) and G(C, D) are isomorphic if and only if n = m and there exists P ∈ GL n (k) such that either (C, D) = (P t AP, P −1 BP −1t ) or (C, D) = (P t B −1 P, P −1 A −1 P −1t )

doi:10.4171/jncg/150
fatcat:dkdzgigdhze5zhetjmp5mzqose