Equivariant deformation cohomology and group actions on compatible Hom-Leibniz algebras

Purpose This paper introduces and studies compatible G-Hom-Leibniz algebras, namely compatible Hom-Leibniz algebras equipped with a finite group action. The aim is to develop their equivariant deformation cohomology theory and to analyze how this cohomology governs equivariant formal, finite-order and infinitesimal deformations. The paper also examines the role of G-Nijenhuis operators in generating trivial equivariant infinitesimal deformations. Design/methodology/approach We construct a graded Lie algebra whose Maurer–Cartan elements encode compatible G-Hom-Leibniz algebra structures. Using this graded Lie algebra, we define an equivariant cohomology theory with coefficients in the algebra itself and apply it to study several types of equivariant deformations. We further introduce the notion of a G-Nijenhuis operator and explore its connection with trivial infinitesimal deformations. Findings The proposed graded Lie algebra completely characterizes compatible G-HomLeibniz algebras. The resulting equivariant cohomology controls their deformation behaviour and provides criteria for extending finite-order deformations. We also show that trivial equivariant infinitesimal deformations correspond precisely to G-Nijenhuis operators. Originality/value This work provides the first systematic treatment of compatible Hom-Leibniz algebras under group actions and develops their equivariant deformation cohomology. The introduction of G-Nijenhuis operators and their deformation-theoretic interpretation offers new tools for further research in Hom-type algebras and related areas.