Constructions of Quadratic \(n\) -ary Hom-Nambu Algebras
The aim of this paper is provide a survey on \(n\) -ary Hom-Nambu algebras and study quadratic \(n\) -ary Hom-Nambu algebras, which are \(n\) -ary Hom-Nambu algebras with an invariant, nondegenerate and symmetric bilinear forms that are also \(\alpha \) -
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Abstract The aim of this paper is provide a survey on n-ary Hom-Nambu algebras and study quadratic n-ary Hom-Nambu algebras, which are n-ary Hom-Nambu algebras with an invariant, nondegenerate and symmetric bilinear forms that are also α-symmetric and β-invariant where α and β are twisting maps. We provide various constructions of quadratic n-ary Hom-Nambu algebras. Also is discussed their connections with representation theory and centroids.
1 Introduction The main motivations to study n-ary algebras came firstly from Nambu mechanics [34] where a ternary bracket allows to use more than one hamiltonian and recently from string theory and M-branes which involve naturally an algebra with ternary operation called Bagger-Lambert algebra [11]. Also ternary operations appeared in the study of some quarks models see [22–24]. For more general theory and further results see references [5, 6, 15–17, 20, 21, 26, 27, 35, 38]. Algebras endowed with invariant nondegenerate symmetric bilinear form (scalar product) appeared also naturally in several domains in mathematics and physics. Such algebras were intensively studied for binary Lie and associative algebras. The main results are that called double extension given by Medina and Revoy [33] and T*-extension given by Bordemann [14]. These fundamental results were extended to F. Ammar · S. Mabrouk Faculté des Sciences, Université de Sfax, Sfax, Tunisia e-mail: [email protected] S. Mabrouk e-mail: [email protected] A. Makhlouf (B) Université de Haute-Alsace, 4 rue des Frères Lumière, 68093 Mulhouse, France e-mail: [email protected] A. Makhlouf et al. (eds.), Algebra, Geometry and Mathematical Physics, 201 Springer Proceedings in Mathematics & Statistics 85, DOI: 10.1007/978-3-642-55361-5_12, © Springer-Verlag Berlin Heidelberg 2014
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n-ary algebras in [18]. The extension to Hom-setting for binary case was introduced and studied in [13]. For further results about Hom-type algebras, see refs [2–4, 25, 28, 30–32, 41, 42]. In this paper we summarize in the Sect. 2 definitions of n-ary Hom-Nambu algebras and recall the constructions using twisting principles and tensor product with n-ary algebras of Hom-associative type. In Sect. 3, we introduce the notion of quadratic n-ary Hom-Nambu algebra, generalizing the notion introduced for binary Hom-Lie algebras in [13]. A more general notion called Hom-quadratic n-ary HomNambu algebra is introduced by twisting the invariance identity. In Sect. 4, we show that a quadratic n-ary Hom-Nambu algebra gives rise to a quadratic Hom-Leibniz algebra. A connection with representation theory is discussed in Sect. 5. We deal in particular with adjoint and coadjoint representations, extending the representation theory initiated in [13, 36]. Several procedures to built quadratic n-ary Hom-Nambu algebras are provided in Sect. 6. We use twisting principles, tensor product and T*extension to construct quadratic n-ary Hom-Nambu algebras. Moreover we show that one may derive from quadratic n-ary Hom-Nambu algebra ones of
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