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Investigates the cohomology and deformation theory of compatible 3-Lie superalgebras, suggesting new insights into their algebraic structure.
In this paper, we first introduce the notions of a compatible 3-Lie superalgebras and its representation. We construct a bidifferential graded Lie algebra whose Maurer–Cartan elements are compatible 3-Lie superalgebras. We also obtain the bidifferential graded Lie algebra which controls deformations of a compatible 3-Lie superalgebra. Then we investigate the cohomology theory of compatible 3-Lie superalgebras and consider the connection between the cohomology group of compatible 3-Lie superalgebras and the cohomology group of 3-Lie superalgebras. Furthermore, we develop the 1-parameter formal deformation theory of compatible 3-Lie superalgebras and prove that it is governed by the cohomology groups. At last, we study abelian extensions of compatible 3-Lie superalgebras and classify them by the second cohomology group.
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Guan et al. (2026) studied this question.
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