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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.10949 |
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Table of Contents:
- This paper studies the associated Levi-Civita products of a Leibniz algebra with a nondegenerate skew-symmetric $2$-cocycle. Such products form into the notion of an anti-pre-Leibniz algebra, which is characterized as a Leibniz-admissible algebra which renders a representation of the sub-adjacent Leibniz algebra through the negative multiplication operators. Such a characterization serves as the converse side of the role that pre-Liebniz algebras play in the splitting theory of Leibniz algebras, which justifies the name of anti-pre-Leibniz algebras. There is a compatible anti-pre-Leibniz algebra structure on a Leibniz algebra if and only if there is an invertible anti-$\mathcal{O}$-operator of the Leibniz algebra. Another important role that anti-pre-Leibniz algebras play is that they give a new characterization of Novikov dialgebras, that is, a Novikov dialgebra is interpreted as a transformed pre-Leibniz algebra which gives rise to an anti-pre-Leibniz algebra structure through specific combinations of multiplications. The properties of Novikov dialgebras are also further investigated.