Nambu-Lie Algebras Research Articles

3-Lie algebras, ternary Nambu-Lie algebras and the Yang-Baxter equation

We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, 3-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple 3-Lie algebras. We show that the Yang-Baxter operators obtained are not gauge equivalent to the transposition operator, and we consider the problem of deforming the operators to obtain new solutions to the Yang-Baxter equation. We discuss the applications of this deformation procedure to the construction of (framed) link invariants.

On n-ary Hom–Lie H-pseudoalgebras

The aim of this paper is to study n-ary Hom–Lie H-pseudoalgebras generalizing both n-ary Hom–Nambu–Lie algebras and n-Lie H-pseudoalgebras. We list examples of the new structure and present some properties and the construction theorem. Furthermore, we provide a way to construct an n-ary Hom–Nambu–Lie algebra from an n-ary Hom–Lie H-pseudoalgebra. The equivalent definition of n-ary Hom–Lie H-pseudoalgebra is also described. Finally, we discuss the prospects of n-ary Hom–Lie H-pseudoalgebras in physics.

Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras

As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.

Representations and cohomology of [formula omitted]-ary multiplicative Hom–Nambu–Lie algebras

The aim of this paper is to provide cohomologies of n -ary Hom–Nambu–Lie algebras governing central extensions and one parameter formal deformations. We generalize to n -ary algebras the notions of derivation and representation introduced by Sheng for Hom–Lie algebras. Also we show that a cohomology of n -ary Hom–Nambu–Lie algebras could be derived from the cohomology of Hom–Leibniz algebras.

Ternary q-Virasoro–Witt Hom–Nambu–Lie algebras

In this paper we construct ternary q-Virasoro–Witt algebras which q-deform the ternary Virasoro–Witt algebras constructed by Curtright, Fairlie and Zachos using su(1, 1) enveloping algebra techniques. The ternary Virasoro–Witt algebras constructed by Curtright, Fairlie and Zachos depend on a parameter and are not Nambu–Lie algebras for all but finitely many values of this parameter. For the parameter values for which the ternary Virasoro–Witt algebras are Nambu–Lie, the corresponding ternary q-Virasoro–Witt algebras constructed in this paper are also Hom–Nambu–Lie because they are obtained from the ternary Nambu–Lie algebras using the composition method. For other parameter values this composition method does not yield a Hom–Nambu–Lie algebra structure for q-Virasoro–Witt algebras. We show however, using a different construction, that the ternary Virasoro–Witt algebras of Curtright, Fairlie and Zachos, as well as the general ternary q-Virasoro–Witt algebras we construct, carry a structure of the ternary Hom–Nambu–Lie algebra for all values of the involved parameters.

Ternary Hom–Nambu–Lie algebras induced by Hom–Lie algebras

The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom–Nambu–Lie algebras obtained using this construction.

Generalization of n-ary Nambu algebras and beyond

The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu–Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom–Nambu–Lie structures with diagonal homomorphism.

Notes on cohomologies of ternary algebras of associative type

The aim of this paper is to investigate the cohomologies for ternary algebras of associative type.We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan's construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a usual deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad.

Notes on cohomologies of ternary algebras of associative type

The aim of this paper is to investigate the cohomologies for ternary algebras of associative type. We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan's construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad.

Graded multiple analogs of Lie algebras

Graded analogs of (n,k,r)-Lie algebras (in particular, of Nambu–Lie algebras), introduced by the authors, are defined and their general property are studied.

Linearization of Nambu Structures

Nambu structures are a generalization of Poisson structures in Hamiltonian dynamics, and it has been shown recently by several authors that, outside singular points, these structures are locally an exterior product of commuting vector fields. Nambu structures also give rise to co-Nambu differential forms, which are a natural generalization of integrable 1-forms to higher orders. This work is devoted to the study of Nambu tensors and co-Nambu forms near singular points. In particular, we give a classification of linear Nambu structures (integral finite-dimensional Nambu-Lie algebras), and a linearization of Nambu tensors and co-Nambu forms, under the nondegeneracy condition.

Simple Facts Concerning Nambu Algebras

A class of substitution equations arising in the extension of Jacobi identity for $n$-gebras is studied and solved. Graded bracket and cohomology adapted to the study of formal deformations are presented. New identities in the case of Nambu-Lie algebras are proved. The triviality in the Gerstenhaber sense of certain deformed n-skew-symmetric brackets, satisfying the Leibniz rule with respect to a star-product, is shown for n≥ 3.