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Register a new userGeneralized Landau-Lifshitz systems and Lie algebras associated with higher genus curves
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The Wahlquist-Estabrook prolongation method allows to obtain for some PDEs a Lie algebra that is responsible for Lax pairs and Backlund transformations of certain type. We study the Wahlquist-Estabrook algebra of the n-dimensional generalization of the Landau-Lifshitz equation and construct an epimorphism from this algebra onto an infinite-dimensional quasigraded Lie algebra L(n) of certain matrix-valued functions on an algebraic curve of genus 1+(n-3)2^{n-2}. For n=3,4,5 we prove that the Wahlquist-Estabrook algebra is isomorphic to the direct sum of L(n) and a 2-dimensional abelian Lie algebra. Using these results, for any n a new family of Miura type transformations (differential substitutions) parametrized by points of the above mentioned curve is constructed. As a by-product, we obtain a representation of L(n) in terms of a finite number of generators and relations, which may be of independent interest.
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The Wahlquist-Estabrook prolongation method constructs for some PDEs a Lie algebra that is responsible for Lax pairs and Backlund transformations of certain type. We present some general properties of Wahlquist-Estabrook algebras for (1+1)-dimensional evolution PDEs and compute this algebra for the n-component Landau-Lifshitz system of Golubchik and Sokolov for any $nge 3$. We prove that the resulting algebra is isomorphic to the direct sum of a 2-dimensional abelian Lie algebra and an infinite-dimensional Lie algebra L(n) of certain matrix-valued functions on an algebraic curve of genus $1+(n-3)2^{n-2}$. This curve was used by Golubchik, Sokolov, Skrypnyk, Holod in constructions of Lax pairs. Also, we find a presentation for the algebra L(n) in terms of a finite number of generators and relations. These results help to obtain a partial answer to the problem of classification of multicomponent Landau-Lifshitz systems with respect to Backlund transformations. Furthermore, we construct a family of integrable evolution PDEs that are connected with the n-component Landau-Lifshitz system by Miura type transformations parametrized by the above-mentioned curve. Some solutions of these PDEs are described.
In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_infty$-algebra, whose Maurer-Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie $3$-algebra whose Maurer-Cartan elements are O-operators (also called relative Rota-Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of twilled 3-Lie algebras, which we call strict twilled 3-Lie algebras. Finally, we use O-operators to construct explicit twilled 3-Lie algebras, and explain why an $r$-matrix for a 3-Lie algebra can not give rise to a double construction 3-Lie bialgebra. Examples of twilled 3-Lie algebras are given to illustrate the various interesting phenomenon.
In this paper, we introduce the definition of generalized BiHom-Lie algebras and generalized BiHom-Lie admissible algebras in the category ${}_H{mathcal M}$ of left modules for any quasitriangular Hopf algebra $(H, R) $. Also, we describe the BiHom-Lie ideal structures of the BiHom-associative algebras.
Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton equations too. In particular, such deformations for the three-dimensional and infinite-dimensional algebras are described by the Boussinesq equation and KP hierarchy, respectively.
We introduce the conception of matched pairs of $(H, beta)$-Lie algebras, construct an $(H, beta)$-Lie algebra through them. We prove that the cocycle twist of a matched pair of $(H, beta)$-Lie algebras can also be matched.
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