On Representations of Quantum Conjugacy Classes of GL(n) |
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Authors: | Thomas Ashton Andrey Mudrov |
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Institution: | 1. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK
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Abstract: | Let O be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by T the maximal torus of diagonal matrices in GL(n). With every ${a \in O \cap T}$ we associate a highest weight module M a over the quantum group ${U_q \bigl(\mathfrak{g} \mathfrak{l}(n)\bigr)}$ and an equivariant quantization ${\mathbb{C}_{\hbar,a}O]}$ of the polynomial ring ${\mathbb{C}O]}$ realized by operators on M a . All quantizations ${\mathbb{C}_{\hbar,a}O]}$ are isomorphic and can be regarded as different exact representations of the same algebra, ${\mathbb{C}_{\hbar}O]}$ . Similar results are obtained for semisimple adjoint orbits in ${\mathfrak{g} \mathfrak{l}(n)}$ equipped with the canonical GL(n)-invariant Poisson structure. |
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