On split involutive regular BiHom-Lie superalgebras

Abstract The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra L {\mathfrak{L}} is of the form L = U + ∑ α I α {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of L {\mathfrak{L}} , satisfying [I α , I β ] = 0 if [α] ≠ [β]. In the case of L {\mathfrak{L}} being of maximal length, the simplicity of L {\mathfrak{L}} is also characterized in terms of connections of roots.


Introduction
The notion of Hom-Lie algebras was first introduced by Hartwig, Larsson and Silvestrov in [1], who developed an approach to deformations of the Witt and Virasoro algebras based on σ-deformations. In fact, Hom-Lie algebras include Lie algebras as a subclass, but the deformation of Lie algebras is twisted by a homomorphism.
A BiHom-algebra is an algebra in which the identities defining the structure are twisted by two homomorphisms ϕ and ψ. This class of algebras was introduced from a categorical approach in [2] which can be viewed as an extension of the class of Hom-algebras. If the two linear maps are the same automorphisms, BiHom-algebras will return to Hom-algebras. These algebraic structures include BiHomassociative algebras, BiHom-Lie algebras and BiHom-bialgebras. The representation theory of BiHom-Lie algebras was introduced by Cheng and Qi in [3], in which BiHom-cochain complexes, derivations, central extensions, derivation extensions, trivial representations and adjoint representations of BiHom-Lie algebras were studied. More applications of BiHom-algebras, BiHom-Lie superalgebras, BiHom-Lie color algebras and BiHom-Novikov algebras can be found in [4][5][6][7].
The class of the split algebras is especially related to addition quantum numbers, graded contractions and deformations. For instance, for a physical system which displays a symmetry, it is interesting to know the detailed structure of the split decomposition, since its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such system. Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics.
This paper is organized as follows. In Section 3, we prove that such an arbitrary involutive regular BiHom-Lie superalgebra L is of the form = + ∑ U I α α L with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of L, satisfying [I α , . In Section 4, under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized.

Preliminaries
Throughout this paper, we will denote by the set of all nonnegative integers and by the set of all integers. Split involutive regular BiHom-Lie superalgebras are considered of arbitrary dimension and over an arbitrary base field . Now we recall some basic definitions and results related to our paper from [11] and [12].
Note that 0 L is a BiHom-Lie algebra. The usual regularity concepts will be understood in the graded sense. That is, an involutive super-subalgebra

L
. We call that L is a split involutive regular BiHom-Lie superalgebra with respect to H if We also say that Λ is the root system of L.
The following two lemmas are analogous to the results of [11] and [12].
be a split involutive regular BiHom-Lie superalgebra. Then, for any Lemma 2.3. The following assertions hold:

Decompositions
In what follows, L denotes a split involutive regular BiHom-Lie superalgebra and is the corresponding root-space decomposition. We begin by developing the techniques of connections of roots in this section.
Definition 3.1. Let α and β be two nonzero roots. We will say that α is connected to β if either We will also say that {α 1 ,…,α k } is a connection from α to β.
The proof of the next result is analogous to the one in [12].
Proposition 3.2. The relation ∼ in Λ, defined by α ∼ β if and only if α is connected to β, is an equivalence relation.
By Proposition 3.2, we can consider the quotient set being the set of nonzero roots which are connected to α. Our next goal is to associate an ideal the direct sum of the two subspaces above: For any [α] ∈ Λ/∼, the following assertions hold:

Example
In this section, we provide an example to clarify the results in Section 3, generalizing the example of [23].
In [24], Rittenberg and Wyler introduced the definition of ×  Suppose that L is a direct sum of graded components, If L satisfies skew-symmetry and Jacobi identity, then L is referred to as a × 2 2 -graded Lie superalgebra.
In [25], denote by e i,j the matrix with zero everywhere except a 1 on position (i, j), where the row and the column indices run from 1 to 2m + 2n +1. We introduce the following elements: Tolstoy proved that the × Using Theorem 2.7 in [5], we can construct an involutive BiHom-Lie superalgebra from the Lie superalgebra pso(2m + 1, 2n).