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<p>A non-associative superalgebra is called <i>pre-symplectic</i> if it is equipped with a non-degenerate, anti-symmetric bilinear form. It is called <i>quasi-Frobenius</i> if, in addition, it is a Lie superalgebra and the form is closed. We introduce the Levi-Civita product associated with pre-symplectic superalgebras and establish its existence and uniqueness. We then introduce the symplectic product associated with quasi-Frobenius Lie superalgebras. We prove that while such a product always exists, it is not unique. We therefore define a <i>natural symplectic product</i> that depends only on the Lie structure and the bilinear form. When the curvature of this product vanishes, the superalgebra is called a <i>flat quasi-Frobenius Lie superalgebra</i>. In this paper, we study flat quasi-Frobenius Lie superalgebras and introduce the notion of a <i>flat double extension</i>. We prove that the double extension process characterizes such superalgebras. More precisely, every flat orthosymplectic (resp. periplectic) quasi-Frobenius Lie superalgebra can be obtained by a sequence of flat double extensions starting from an abelian one (resp. the trivial one). Moreover, we show that every flat quasi-Frobenius Lie superalgebra is nilpotent with a degenerate center. We apply our results to obtain a complete classification of such superalgebras of total dimension at most 5.</p>

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Flat quasi-Frobenius Lie superalgebras

  • 期刊论文

  • 发表日期：

    2026年4月6日

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  • Sofiane Bouarroudj,
  • Hamza El Ouali

摘要

A non-associative superalgebra is called pre-symplectic if it is equipped with a non-degenerate, anti-symmetric bilinear form. It is called quasi-Frobenius if, in addition, it is a Lie superalgebra and the form is closed. We introduce the Levi-Civita product associated with pre-symplectic superalgebras and establish its existence and uniqueness. We then introduce the symplectic product associated with quasi-Frobenius Lie superalgebras. We prove that while such a product always exists, it is not unique. We therefore define a natural symplectic product that depends only on the Lie structure and the bilinear form. When the curvature of this product vanishes, the superalgebra is called a flat quasi-Frobenius Lie superalgebra. In this paper, we study flat quasi-Frobenius Lie superalgebras and introduce the notion of a flat double extension. We prove that the double extension process characterizes such superalgebras. More precisely, every flat orthosymplectic (resp. periplectic) quasi-Frobenius Lie superalgebra can be obtained by a sequence of flat double extensions starting from an abelian one (resp. the trivial one). Moreover, we show that every flat quasi-Frobenius Lie superalgebra is nilpotent with a degenerate center. We apply our results to obtain a complete classification of such superalgebras of total dimension at most 5.