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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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From tunnels to towers: Quantum scars from Lie algebras and q -deformed Lie algebras
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Abstract For each prime 𝑝 and each positive integer 𝑑, we construct the first examples of second countable, topologically simple 𝑝-adic Lie groups of dimension 𝑑 whose Lie algebras are abelian.This answers several questions of Glöckner and Caprace–Monod.The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1103/PhysRevResearch.2.043305
