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Idempotents de Jones-Wenzl évaluables aux racines de l'unité et représentation modulaire sur le centre de U¯_{q}sl(2)

Abstract : Let p in N^*. We define a family of idempotents (and nilpotents) in the Temperley-Lieb algebras at 4p-th roots of unity which generalizes the usual Jones-Wenzl idempotents. These new idempotents correspond to finite dimentional simple and projective indecomposable representations of the restricted quantum group U¯_{q}sl(2), where q is a 2p-th root of unity. In the manner of the [BHMV95] topological quantum field theorie (TQFT), they provide a canonical basis in colored skein modules to define mapping class groups representations. In the torus case, this basis allows us to obtain a partial match between the negative twist and the buckling actions, and the [LM94] induced representation of SL₂(ℤ) on the center of U¯_{q}sl(2), which extends non trivially the [RT91] representation of SL₂(ℤ).
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Elsa Ibanez. Idempotents de Jones-Wenzl évaluables aux racines de l'unité et représentation modulaire sur le centre de U¯_{q}sl(2). Mathématiques générales [math.GM]. Université Montpellier, 2015. Français. ⟨NNT : 2015MONTS233⟩. ⟨tel-02067407⟩

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