Automorphic Lie algebras and corresponding integrable systems

Bury, Rhys T.; Михайлов, А. В.

doi:10.1016/j.difgeo.2020.101710

Cited by 8 publications

(16 citation statements)

References 15 publications

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“…Including the information of the determinant of invariant vectors yields a partial description of ω 2 , enough to describe the Killing form of the Automorphic Lie Algebra and its abelianisation (as well as κ i = 1 2 α∈Φ ω 2 (α, −α)). Moreover, it shows that ω 2 can only take two values, thereby reproducing isomorphisms obtained in [27,28,29,2,22,24]. The challenge remains to find the complete description of ω 2 , and hence the Automorphic Lie Algebra, from the embedding of the reduction group into Aut(g).…”

Section: Introductionmentioning

confidence: 66%

Hereditary automorphic Lie algebras

Knibbeler

Lombardo

Sanders

2019

Commun. Contemp. Math.

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We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

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Section: Introductionmentioning

confidence: 66%

Hereditary automorphic Lie algebras

Knibbeler

Lombardo

Sanders

2019

Commun. Contemp. Math.

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“…In Section 4 we will present two examples of new NLEE whose Lax representation conform with case (iv) of (3.2). Detailed analysis of the realizations of D h as reduction groups of Lax pairs has been performed by Mikhailov, Lombardo and Bury [43,44,42,45,8,9]. It depends substantially not only on the realization of r(λ), but also on the choice of the orbit of D h .…”

Section: 2mentioning

confidence: 99%

“…The situation changes if we consider the cases (ii) and (iv) in (3.2). Then the Lax pair becomes polynomial in λ and λ −1 , see [8,43]. Now we must use two asymptotic expansions of ξ k (x, t, λ): one for |λ| ≫ 1 and another valid for |λ| ≪ 1:…”

Section: 2mentioning

confidence: 99%

“…The scope of the present paper is limited to the Z h and D h reduction groups. Of course there are quite a few examples of NLEE related to the tetrahedral T, octahedral O and even to the icosahedral Y reduction groups, see [8] and the references therein. The deeper studies of these NLEE should be based on the relevant automorphic Lie algebras A(g) [43,44,45,42].…”

Section: Vladimir S Gerdjikov Rossen I Ivanov and Aleksander A Stefanovmentioning

confidence: 99%

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Riemann-Hilbert problem, integrability and reductions

Gerdjikov¹,

Ivanov²,

Stefanov³

2019

Journal of Geometric Mechanics

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The present paper is dedicated to integrable models with Mikhailov reduction groups G R ≃ D h . Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the G R -action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with D h symmetries are presented. 2010 Mathematics Subject Classification. Primary: 35K10, 35Q15; Secondary: 37K15, 35Q55.

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“…The classification of automorphic Lie algebras is part of the programme of classification of Lax operators and hence of integrable systems. The problem of classification of automorphic Lie algebras corresponding to finite reduction groups had been extensively studied in [14] and independently in [15]. An alternative approach to automorphic Lie algebras and further development can be found in [16], [17].…”

Section: Introductionmentioning

confidence: 99%