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Automorphic Lie algebras and corresponding integrable systems. (English) Zbl 1498.17049

Summary: We study automorphic Lie algebras and their applications to integrable systems. Automorphic Lie algebras are a natural generalisation of celebrated Kac-Moody algebras to the case when the group of automorphisms is not cyclic. They are infinite dimensional and almost graded. We formulate the concept of a graded isomorphism and classify \(sl(2,\mathbb{C})\) based automorphic Lie algebras corresponding to all finite reduction groups. We show that hierarchies of integrable systems, their Lax representations and master symmetries can be naturally formulated in terms of automorphic Lie algebras.

MSC:

17B80 Applications of Lie algebras and superalgebras to integrable systems
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20G45 Applications of linear algebraic groups to the sciences
37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
53A31 Differential geometry of submanifolds of Möbius space
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