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The aim of this work is to introduce representations of BiHom-left-symmetric algebras and develop its cohomology theory. As applications, we study linear deformations of BiHom-left-symmetric algebras, which are characterized by its second cohomology group with the coefficients in the adjoint representation. The notion of a Nijenhuis operator on a BiHom-left-symmetric algebra is introduced. We will prove that it can generate trivial linear deformations of a BiHom-left-symmetric algebra. [ABSTRACT FROM AUTHOR]

We contribute to the study of Rota–Baxter operators on types of algebras other than associative and Lie algebras. If A is an algebra of a certain type and R is a Rota–Baxter operator on A, one can define a new multiplication on A by means of R and the previous multiplication and ask under what circumstances the new algebra is of the same type as A. Our first main result deals with such a situation in the case of BiHom-Lie algebras. Our second main result is a BiHom analogue of Aguiar's theorem that shows how to obtain a pre-Lie algebra from a Rota–Baxter operator of weight zero on a Lie algebra. The BiHom analogue does not work for BiHom-Lie algebras, but for a new concept we introduce here, called left BiHom-Lie algebra, at which we arrived by defining first the BiHom version of Leibniz algebras. [ABSTRACT FROM AUTHOR]

BiHom-Novikov algebra is a generalized Hom-Novikov algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show that two classes of BiHom-Novikov algebras can be constructed from BiHom-commutative algebras together with derivations and BiHom-Novikov algebras with Rota–Baxter operators, respectively. We show that quadratic BiHom-Novikov algebras are associative algebras and the sub-adjacent BiHom-Lie algebras of BiHom-Novikov algebras are 2-step nilpotent. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov algebras. [ABSTRACT FROM AUTHOR]

We introduce the class of split regular BiHom-Lie superalgebras as the natural extension of the one of split Hom-Lie superalgebras and the one of split Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Lie superalgebra L is of the form L = U + ∑ [ α ] ∈ Λ ∕ ∼ I [ α ] with U a subspace of the Abelian (graded) subalgebra H and any I [ α ] , a well described (graded) ideal of L , satisfying [ I [ α ] , I [ β ] ] = 0 if [ α ] ≠ [ β ] . Under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized and it is shown that L is the direct sum of the family of its simple (graded) ideals. [ABSTRACT FROM AUTHOR]

We introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. We show that an arbitrary split regular BiHom-Lie algebra L is of the form L = U + ∑ j I j with U a linear subspace of a fixed maximal abelian subalgebra H and any I j a well described (split) ideal of L , satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals. [ABSTRACT FROM AUTHOR]