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Based on Nijenhuis-Richardson bracket and bidegree on the cohomology complex for a Lie conformal algebra, we develop a twisting theory of Lie conformal algebras. By using derived bracket constructions, we construct L ∞ -algebras from (quasi-)twilled Lie conformal algebras. And we show that the result of the twisting by a C [ ∂ ] -module homomorphism on a (quasi-)twilled Lie conformal algebra is also a (quasi-)twilled Lie conformal algebra if and only if the C [ ∂ ] -module homomorphism is a Maurer-Cartan element of the L ∞ -algebra. In particular, we show that relative Rota-Baxter type operators on Lie conformal algebras are Maurer-Cartan elements. Besides, we propose a new algebraic structure, called NS-Lie conformal algebras, that is closely related to twisted relative Rota-Baxter operators and Nijenhuis operators on Lie conformal algebras. As an application of twisting theory, we give the cohomology of twisted relative Rota-Baxter operators and study their deformations. [ABSTRACT FROM AUTHOR]

In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce N -structures on bimodules over pre-Lie algebras. We show that an N -structure gives rise to a hierarchy of pairwise compatible -operators. We study solutions of the strong Maurer–Cartan equation on the twilled pre-Lie algebra associated to an -operator, which gives rise to a pair of N -structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra are corresponding to N -structures on the bimodule ( ∗ ; ad ∗ , − R ∗) , and KVB-structures are corresponding to solutions of the strong Maurer–Cartan equation on a twilled pre-Lie algebra associated to an -matrix. [ABSTRACT FROM AUTHOR]

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization–depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang–Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang–Baxter equation and the associative Yang–Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang–Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang–Baxter equation gives an adm-Poisson bialgebra. Finally, the notions of an -operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang–Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar. [ABSTRACT FROM AUTHOR]

The notion of an F -manifold algebra is the underlying algebraic structure of an F -manifold. We introduce the notion of pre-Lie formal deformations of commutative associative algebras and show that F -manifold algebras are the corresponding semi-classical limits. We study pre-Lie infinitesimal deformations and extension of pre-Lie n -deformation to pre-Lie (n + 1) -deformation of a commutative associative algebra through the cohomology groups of pre-Lie algebras. We introduce the notions of pre- F -manifold algebras and dual pre- F -manifold algebras, and show that a pre- F -manifold algebra gives rise to an F -manifold algebra through the sub-adjacent associative algebra and the sub-adjacent Lie algebra. We use Rota-Baxter operators, more generally O -operators and average operators on F -manifold algebras to construct pre- F -manifold algebras and dual pre- F -manifold algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study non-abelian extensions of Leibniz algebras using two different approaches. First we construct two Leibniz 2-algebras using biderivations of Leibniz algebras and show that under a condition on centers, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. Furthermore, under this condition, non-abelian extensions are classified by homotopy classes of Leibniz 2-algebra morphisms. Then we give a description of non-abelian extensions of Leibniz algebras in terms of Maurer-Cartan elements in a differential graded Lie algebra. [ABSTRACT FROM AUTHOR]

The statement in [2, Example 3.6] is not true in general. [ABSTRACT FROM AUTHOR]

A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from r -matrices through Lie algebra actions. We study cohomologies of pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart algebras are classified by the second cohomology groups. We introduce the notion of crossed modules for pre-Lie-Rinehart algebras and show that they are classified by the third cohomology groups of pre-Lie-Rinehart algebras. At last, we use (pre-)Lie-Rinehart 2-algebras to characterize the crossed modules for (pre-)Lie Rinehart algebras. [ABSTRACT FROM AUTHOR]

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between 𝒪 -operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator "connects" two 𝒪 -operators on a pre-Lie algebra whose any linear combination is still an 𝒪 -operator in certain sense and hence compatible L -dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras. [ABSTRACT FROM AUTHOR]