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arXiv · 2506.23710

A New Definition of Superbiderivations for Lie Superalgebras

Alfonso Di Bartolo·Francesco Paolo Di Fatta·Gianmarco La Rosa
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Abstract

In this paper, we study superbiderivations on Lie superalgebras from structural and geometric perspectives. Motivated by the classical fact that the bracket of a Lie algebra is itself a biderivation, we propose a new definition of superbiderivation for Lie superalgebras, one that requires the bracket to be a superbiderivation, a condition not satisfied by existing definitions in the literature. Our focus is on complete Lie superalgebras, a natural generalization of semisimple Lie algebras that has emerged as a promising framework in the search for alternative structural notions. In this setting, we introduce and study linear supercommuting maps, comparing our definition with previous proposals. Finally, we present two applications: one involving the superalgebra of superderivations of the Heisenberg Lie superalgebra and another offering initial geometric insights into deformation theory via superbiderivations.

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Alfonso Di Bartolo, Francesco Paolo Di Fatta, Gianmarco La Rosa. 2025-06-30. A New Definition of Superbiderivations for Lie Superalgebras. https://arxiv.org/abs/2506.23710

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