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1. Almost PI algebras are PI

   https://doi.org/10.1090/proc/15818  

   Larsen, Michael; Shalev, Aner (April 2022, Proceedings of the American Mathematical Society)

   We define the notion of an almost polynomial identity of an associative algebra R R , and show that its existence implies the existence of an actual polynomial identity of R R . A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras. 

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2. Darboux transformations of integrable couplings and applications

   Ma, Wen-Xiu; Zhang, Yu-Juan (November 2017, Reviews in mathematical physics)

   A formulation of Darboux transformations is proposed for integrable couplings, based on non-semisimple matrix Lie algebras. Applications to a kind of integrable couplings of the AKNS equations are made, along with an explicit formula for the associated Bäcklund transformation. Exact one-soliton-like solutions are computed for the integrable couplings of the second- and third-order AKNS equations, and a type of reduction is created to generate integrable couplings and their one-soliton-like solutions for the NLS and MKdV equations. 

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3. Tempered Homogeneous Spaces III

   Benoist, Y.; Kobayashi, T. (January 2021, Journal of Lie Theory)

   Abstract. Let 𝘎 be a real semisimple algebraic Lie group and 𝘏 a real reductive algebraic subgroup. We describe the pairs (𝘎,𝘏) for which the representation of 𝘎 in L²(𝘎/𝘏) is tempered. The proof gives the complete list of pairs (𝘎,𝘏) for which L²(𝘎/𝘏) is not tempered. When 𝘎 and 𝘏 are complex Lie groups, the temperedness condition is characterized by the fact that 𝘵𝘩𝘦 𝘴𝘵𝘢𝘣𝘪𝘭𝘪𝘻𝘦𝘳 𝘪𝘯 𝘏 𝘰𝘧 𝘢 𝘨𝘦𝘯𝘦𝘳𝘪𝘤 𝘱𝘰𝘪𝘯𝘵 𝘰𝘯 𝘎/𝘏 𝘪𝘴 𝘷𝘪𝘳𝘵𝘶𝘢𝘭𝘭𝘺 𝘢𝘣𝘦𝘭𝘪𝘢𝘯. 𝘔𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘤𝘴 𝘚𝘶𝘣𝘫𝘦𝘤𝘵 𝘊𝘭𝘢𝘴𝘴𝘪𝘧𝘪𝘤𝘢𝘵𝘪𝘰𝘯: Primary 22𝐄46; secondary 43𝐀85, 22𝐅30. 𝘒𝘦𝘺 𝘞𝘰𝘳𝘥𝘴: Lie groups, homogeneous spaces, tempered representations, unitary representations, matrix coefficients, symmetric spaces. 

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4. Non-Abelian symmetry-resolved entanglement entropy

   https://doi.org/10.21468/SciPostPhys.17.5.127  

   Bianchi, Eugenio; Dona, Pietro; Kumar, Rishabh (January 2024, SciPost Physics)

   We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2) symmetry. 

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5. Mapping class group representations from non-semisimple TQFTs

   https://doi.org/10.1142/S0219199721500917  

   De_Renzi, Marco; Gainutdinov, Azat M; Geer, Nathan; Patureau-Mirand, Bertrand; Runkel, Ingo (February 2023, Communications in Contemporary Mathematics)

   In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category [Formula: see text]. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hep-th/9405167]. Finally, we show that, when [Formula: see text] is the category of finite-dimensional representations of the small quantum group of [Formula: see text], the action of all Dehn twists for surfaces without marked points has infinite order. 

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