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1. Monodromy of the Casimir connection of a symmetrisable Kac–Moody algebra

   https://doi.org/10.1007/s00222-024-01242-8  

   Appel, Andrea; Toledano Laredo, Valerio (May 2024, Inventiones mathematicae)

   Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group BW on V. We then prove that this action is canonically equivalent to the quantum Weyl group action of BW on a quantum deformation of V, that is an integrable, category O module V over the quantum group Uhg such that V/hV is isomorphic to V. This extends a result of the second author which is valid for g semisimple. 

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2. Semisimplification of contragredient Lie algebras

   https://doi.org/10.2140/om.2024.1.211  

   Angiono, Iván; Plavnik, Julia; Sanmarco, Guillermo (January 2025, Orbita Mathematicae)

   We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic p with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category. 

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3. The Galois action on the lower central series of the fundamental group of the Fermat curve

   https://doi.org/10.1007/s11856-023-2571-z  

   Davis, Rachel; Pries, Rachel; Wickelgren, Kirsten (November 2023, Israel Journal of Mathematics)

   Abstract. Information about the absolute Galois group G K of a number field K is encoded in how it acts on the ´etale fundamental group π of a curve X defined over K. In the case that K = Q ( ζ n ) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderson determined the action of G K on the ´etale homology with coefficients in Z/nZ. The ´etale homology is the first quotient in the lower central series of the ´etale fundamental group. In this paper, we determine the Galois module structure of the graded Lie algebra for π. As a consequence, this determines the action of G K on all degrees of the associated graded quotient of the lower central series of the ´etale fundamental group of the Fermat curve of degree n, with coefficients in Z/nZ. 

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4. Hopf algebra actions in tensor categories

   https://doi.org/10.1007/s00031-020-09560-w  

   Bischoff, M.; Davydov, A. (April 2020, Transformation Groups)

   We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the representation category of the acting group. 

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5. Cohomology of finite tensor categories: Duality and Drinfeld centers

   https://doi.org/10.1090/tran/8548  

   Negron, Cris; Plavnik, Julia (March 2022, Transactions of the American Mathematical Society)

   We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. 

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