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arXiv · 2608.30182

On Completeness of $n$-ary Hom-Nambu and Hom-Lie Superalgebras

Mohammad Reza Farhangdoost·Mohammad Reza Hafezi·Sergei Silvestrov
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Abstract

We introduce and study completeness for multiplicative $n$-ary Hom-Nambu superalgebras. Because an $n$-ary Hom-Nambu bracket is not necessarily totally super-skew-symmetric, we define its center as the intersection of its positional centers. A multiplicative $n$-ary Hom-Nambu superalgebra is complete when its center is trivial, and every $\alpha^{k+1}$-derivation is inner for all $k \geq 0$. We show that this notion reduces to the usual completeness for multiplicative $n$-Hom-Lie superalgebras. Furthermore, we establish a completeness criterion for surjective brackets with a zero twisting map and provide a complete $n$-ary Hom-Nambu superalgebra that is not an $n$-Hom-Lie superalgebra. We also study the direct sums of complete $n$-Hom-Lie superalgebras and the behavior of centers under twisting. Finally, starting from a multiplicative Hom-Lie superalgebra, we consider recursively induced multiplicative $n$-ary Hom-Nambu superalgebras. When the twisting map is surjective, we prove that triviality of the center is preserved and reflected in this construction. We also observe that every binary $\alpha^k$-derivation satisfies the corresponding relative $\alpha^k$-derivation identity for the induced bracket. Finally, we determine the complete members in the selected low-dimensional Hom-Lie and $3$-Hom-Lie superalgebra classifications.

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Mohammad Reza Farhangdoost, Mohammad Reza Hafezi, Sergei Silvestrov. 2026-08-31. On Completeness of $n$-ary Hom-Nambu and Hom-Lie Superalgebras. https://arxiv.org/abs/2608.30182

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