Generalization of n-ary Nambu algebras and beyond

By H. Ataguema, Abdenacer Makhlouf, Sergei Silvestrov. 2008

135 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Astronomy and Astrophysics, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/c04765c897a95c710c78622af30e91bee945c7a0

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Citations in the corpus

Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras (2009, background): “See also [2, 3, 8, 19, 23, 25, 26] for other works on twisted algebraic structures.”
Paradigm of Nonassociative Hom-algebras and Hom-superalgebras (2010, methodology): “This process was introduced in [45] for Lie and associative algebras and more generallyG-associative algebras (see [32] for this class of algebras) and generalized to coalgebras in [35], [33] and to n-ary algebras of Lie and associative types in [5].”
Hom-algebras and homology (2007, background): “e of a Hom-bialgebra [27, Theorem 3.12], generalizing the bialgebra structure on the usual enveloping algebra of a Lie algebra. Besides [21, 26, 27, 30, 31], Hom-algebras have been further studied in [2, 5, 8, 10, 11, 12, 17, 22, 23, 24, 29, 32, 33, 34]. There are two main purposes of this paper: (1”
Rota–Baxter Hom-Lie–Admissible Algebras (2011, background): “Further development could be found in, for example, [34, 56, 57, 7, 9, 56, 15, 46],[68]-[73], and the references therein.”
Hom-Lie Algebras with Symmetric Invariant NonDegenerate Bilinear Forms (2010, unknown): “Further development could be found in [3, 4, 10, 23, 41].”
Representations and cohomology of n-ary multiplicative Hom–Nambu–Lie algebras (2010, methodology): “In this section, we recall the definition of n-ary Hom-Nambu algebras and n-ary HomNambu-Lie algebras, introduced in [6] by Ataguema, Makhlouf and Silvestrov.”
HOM-ALTERNATIVE ALGEBRAS AND HOM-JORDAN ALGEBRAS (2009, methodology): “gebras which were introduced in ( [21, 19]), see also [5, 32, 33, 34, 35, 36]. It is shown in [31] that the universal enveloping Hom-associative algebra carries a structure of Hom-bialgebra. See also [2, 3, 8, 9, 10, 20, 30] for other works on twisted algebraic structures. The purpose of this paper ”
Unital algebras of Hom-associative type and surjective or injective twistings (2009, background): “Also, recently, Ataguema, Makhlouf, and Silvestrov introduced Hom-Nambu and Hom-Nambu-Lie algebras and n-ary versions of Hom-associative algebras in [2], and Makhlouf introduced Hom-Jordan algebras in [14].”
n-ary algebras: a review with applications (2010, background): “The characteristic GJI and FI still admit further generalizations that essentially preserve their original structure, but these (see [17]) will not be considered here.”