The origin of Hom-structures can be found in the physics literature around 1990, appearing in the study of q-deformations of algebras of vector fields, especially Witt and Virasoro algebras, see for instance [1-3]. So far, many authors have studied Hom-type algebras [4-11]. A BiHom-algebra is an algebra in such a way that the identities defining the structure are twisted by two homomorphisms $ \phi $, $ \psi $. The notion of BiHom-Lie algebras was introduced in [12], which is intimately related to both Lie algebras and Hom-Lie algebras. The case of $ \phi = \psi = \mathrm{Id} $ implies BiHom-Lie algebras are Lie algebras and the other case of $ \phi = \psi $ give Hom-Lie algebras. The notion of Lie color algebras was introduced as generalized Lie algebras in 1960 by Ree [13]. In particular, BiHom-Lie color algebras are defined as an extension of BiHom-Lie (super)algebras to $ \Gamma $-graded algebras, where $ \Gamma $ is any abelian group.

As is well-known, the class of the split algebras is specially related to addition quantum numbers, graded contractions and deformations. Recently, the structure of different classes of split algebras have been studied by using techniques of connections of roots (see for instance [14-23]). In the present paper we introduce the class of split BiHom-Lie color algebras of arbitrary dimension as the natural extension of the class of split BiHom-Lie superalgebras studied in [24] and the class of split Lie color algebras studied in [19]. The purpose of this paper is to consider the structure of split regular BiHom-Lie color algebras by the techniques of connections of roots based on some work in [14, 16, 17, 22, 24].

Throughout this paper, split regular BiHom-Lie color algebras $ L $ are considered of arbitrary dimension and over an arbitrary base field $ \mathbb{K} $. This paper is organized as follows. In section 2, we establish the preliminaries on split regular BiHom-Lie color algebras theory. In section 3, we show that such an arbitrary split regular BiHom-Lie color algebra $ L $ with a symmetric root system is of the form $ L = U+\sum_{[\alpha]\in \Lambda/\sim} I_{[\alpha]} $ with $ U $ a subspace of the abelian (graded) subalgebra $ H $ and any $ I_{[\alpha]} $ a well described (graded) ideal of $ L $, satisfying $ [I_{[\alpha]}, I_{[\beta]}] = 0 $ if $ [\alpha]\neq [\beta] $. In section 4, we show that 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.

First we recall the definitions of Lie color algebras and Hom-Lie color algebras. The following definition is well-known from the theory of graded algebra.

Definition 2.1 [10] Let $ \Gamma $ be an abelian group. A bi-character on $ \Gamma $ is a map $ \varepsilon:\Gamma \times \Gamma \rightarrow \mathbb{K}\setminus \{0\} $ satisfying

$ \rm(1) $ $ \varepsilon(\alpha, \beta)\varepsilon(\beta, \alpha) = 1, $

$ \rm(2) $ $ \varepsilon(\alpha, \beta+\gamma) = \varepsilon(\alpha, \beta)\varepsilon(\alpha, \gamma), $

$ \rm(3) $ $ \varepsilon(\alpha+\beta, \gamma) = \varepsilon(\alpha, \gamma)\varepsilon(\beta, \gamma), $

for all $ \alpha, \beta, \gamma \in \Gamma $.

It is clear that $ \varepsilon(\alpha, 0) = \varepsilon(0, \alpha) = 1 $ for any $ \alpha \in \Gamma $, where 0 denotes the identity element of $ \Gamma $.

Definition 2.2 [16] Let $ L = \oplus_{g \in \Gamma}L_{g} $ be a $ \Gamma $-graded $ \mathbb{K} $-vector space. For a nonzero homogeneous element $ v \in L $, denote by $ \bar{v} $ the unique group element in $ \Gamma $ such that $ v \in L_{\bar{v}} $, which will be called the homogeneous degree of $ v $. We shall say that $ L $ is a Lie color algebra if it is endowed with a $ \mathbb{K} $-bilinear map $ [\cdot, \cdot]:L\times L \rightarrow L $ satisfying

$ \rm(1) $ $ [v, w] = -\varepsilon(\bar{v}, \bar{w})[w, v], $ $ (\mathrm{skew} $-$ \mathrm{symmetry}) $

$ \rm(2) $ $ [v, [w, t]] = [[v, w], t]+\varepsilon(\bar{v}, \bar{w})[w, [v, t]], $ $ (\mathrm{Jacobi} $ $ \mathrm{identity}) $

for all homogeneous elements $ v, w, t \in L $.

Lie superalgebras are examples of Lie color algebras with $ \Gamma = \mathbb{Z}_{2} $ and $ \varepsilon(i, j) = (-1)^{ij} $, for any $ i, j \in \mathbb{Z}_{2} $. We also note that $ L_{0} $ is a Lie algebra.

Definition 2.3 [10] A Hom-Lie color algebra is a quadruple $ (L, [\cdot, \cdot], \phi, \varepsilon) $ consisting of a $ \Gamma $-graded $ \mathbb{K} $-vector space $ L $, an even bilinear mapping $ [\cdot, \cdot]: L\times L\rightarrow L $, a homomorphism $ \phi $ and a bi-character $ \varepsilon $ on $ \Gamma $ satisfying

$ \rm(1) $ $ [x, y] = -\varepsilon(\bar{x}, \bar{y})[y, x], $

$ \rm(2) $ $ \varepsilon(\bar{z}, \bar{x})[\phi(x), [y, z]]+\varepsilon(\bar{x}, \bar{y})[\phi(y), [z, x]]+\varepsilon(\bar{y}, \bar{z})[\phi(z), [x, y]] = 0, $

for all homogeneous elements $ x, y, z \in L $, $ \bar{x}, \bar{y}, \bar{z} $ denote the homogeneous degree of $ x, y, z $, respectively. Furthermore, if $ \phi $ is an algebra automorphism, then it is said that $ L $ is a regular Hom-Lie color algebra.

Clearly Hom-Lie algebras and Lie color algebras are examples of Hom-Lie color algebras. Then we recall the definition of BiHom-Lie algebras and give the definition of BiHom-Lie color algebras.

Definition 2.4 [12] A BiHom-Lie algebra over a field $ \mathbb{K} $ is a $ 4 $-tuple $ (L, [\cdot, \cdot], \phi, \psi) $, where $ L $ is a $ \mathbb{K} $-linear space, $ [\cdot, \cdot]:L\times L\rightarrow L $ is a bilinear map and $ \phi, \psi: L\rightarrow L $ are linear mappings satisfying the following conditions$ : $

$ \rm(1) $ $ \phi \circ \psi = \psi \circ \phi $,

$ \rm(2) $ $ [\psi(x), \phi(y)] = -[\psi(y), \phi(x)] $, $ ( $$ \mathrm{BiHom} $-$ \mathrm{skew} $-$ \mathrm{symmetry} $$ ) $

$ \rm(3) $ $ [\psi^{2}(x), [\psi(y), \phi(z)]]+[\psi^{2}(y), [\psi(z), \phi(x)]]+[\psi^{2}(z), [\psi(x), \phi(y)]] = 0 $, $ ( $$ \mathrm{BiHom} $-$ \mathrm{Jacobi} $ $ \mathrm{identity} $$ ) $

for any $ x, y, z \in L $.

Definition 2.5  A BiHom-Lie color algebra $ L $ is a quintuple $ (L, [\cdot, \cdot], \phi, \psi, \varepsilon) $ consisting of a $ \Gamma $-graded space $ L $, an even bilinear mapping $ [\cdot, \cdot]: L\times L\rightarrow L $, two homomorphisms $ \phi, \psi $ and a bi-character $ \varepsilon $ on $ \Gamma $ satisfying

$ \rm(1) $ $ \phi \circ \psi = \psi \circ \phi $,

$ \rm(2) $ $ [\psi(x), \phi(y)] = -\varepsilon(\bar{x}, \bar{y})[\psi(y), \phi(x)] $, $ ( $$ \mathrm{BiHom} $-$ \mathrm{skew} $-$ \mathrm{symmetry} $$ ) $

$ \rm(3) $ $ \varepsilon(\bar{z}, \bar{x})[\psi^{2}(x), [\psi(y), \phi(z)]]+\varepsilon(\bar{x}, \bar{y})[\psi^{2}(y), [\psi(z), \phi(x)]]+\varepsilon(\bar{y}, \bar{z})[\psi^{2}(z), [\psi(x), \phi(y)]] = 0 $, $ ( $$ \mathrm{BiHom} $-$ \mathrm{Jacobi} $ $ \mathrm{identity} $$ ) $

for all homogeneous elements $ x, y, z \in L $, $ \bar{x}, \bar{y}, \bar{z} $ denote the homogeneous degree of $ x, y, z $, respectively. Furthermore, if $ \phi, \psi $ are algebra automorphism, then it is said that $ L $ is a regular BiHom-Lie color algebra.

Lie color algebra are examples of BiHom-Lie color algebras by taking $ \phi = \psi = \mathrm{Id} $. Hom-Lie color algebras are also examples of BiHom-Lie color algebras by considering $ \psi = \phi $.

Example 2.6 Let $ (L, [\cdot, \cdot]) $ be a Lie color algebra, $ \phi, \psi: L\rightarrow L $ two automorphisms and $ \phi \circ \psi = \psi \circ \phi $. If we endow the underlying linear space $ L $ with a new product $ [\cdot, \cdot]^{'}:L\times L\rightarrow L $ defined by $ [x, y]^{'}: = [\phi(x), \psi(y)] $ for any $ x, y \in L $, then we have that $ (L, [\cdot, \cdot]^{'}, \phi, \psi) $ becomes a regular BiHom-Lie color algebra.

Throughout this paper we will consider a regular BiHom-Lie color algebra $ L $ being of arbitrary dimension and over an arbitrary base field $ \mathbb{K} $. $ \mathbb{N} $ denotes the set of all non-negative integers and $ \mathbb{Z} $ denotes the set of all integers. The usual regularity concepts will be understood in the graded sense. For instance, a subalgebra $ A $ of $ L $ is a graded subspace $ A = \oplus_{g \in \Gamma}A_{g} $ such that $ [A, A] \subset A $ and $ \phi(A) = \psi(A) = A $. A graded subspace $ I = \oplus_{g \in \Gamma}I_{g} $ of $ L $ is called an ideal if $ [I, L]+[L, I] \subset I $ and $ \phi(I) = \psi(I) = I $. A BiHom-Lie color algebra $ L $ will be called simple if $ [L, L] \neq 0 $ and its only $ ( $graded$ ) $ ideals are {0} and $ L $.

We introduce the concept of split regular BiHom-Lie color algebra in an analogous way. We begin by considering a maximal abelian graded subalgebra $ H = \oplus_{g \in \Gamma}H_{g} $ among the abelian graded subalgebras of $ L $. We observe that $ H $ is necessarily a maximal abelian subalgebra of $ L $ as the following lemma shows.

Lemma 2.7 Let $ H = \oplus_{g \in \Gamma}H_{g} $ be a maximal abelian graded subalgebra of a BiHom-Lie color algebra $ L $. Then $ H $ is a maximal abelian subalgebra of $ L $.

Proof  We consider an abelian subalgebra $ K $ of $ L $ such that $ H \subset K $. For any $ x \in K $ we have $ [x, H_{g}] = 0 $ for each $ g \in \Gamma $, and so by writing $ x = \sum_{i = 1}^{n} x_{g_{i}} $ with $ x_{g_{i}} \in L_{g_{i}} $ for $ i = 1, \cdots, n $, being $ g_{i} \in \Gamma $ and $ g_{i}\neq g_{j} $ if $ i \neq j $, by the grading we get $ [x_{g_{i}} , H_{g}] = 0 $. Hence, for any $ g_{i} $, $ i = 1, \cdots n $, we have $ (H_{g_{i}} + \mathbb{K}x_{g_{i}} ) \oplus(\oplus_{g \in \Gamma\setminus \{g_{i}\}}H_{g}) $ is an abelian graded subalgebra of $ L $ containing $ H $ and so $ x_{g_{i}} \in H_{g_{i}} $. From here we get $ x \in H $ and then $ K = H $.

Let us introduce the class of split algebras in the framework of regular Lie color algebras $ L $. First, we recall that a Lie color algebra $ (L, [\cdot, \cdot]) $, over a base field $ \mathbb{K} $, is called split respect to a maximal abelian subalgebra $ H $ of $ L $, if $ L $ can be written as the direct sum

$ L = H\oplus (\oplus_{\alpha \in \Delta}L_{\alpha}) $

where

$ L_{\alpha} = \{v_{\alpha}\in L: [h_{0}, v_{\alpha}] = \alpha(h_{0})v_{\alpha}, \ \mathrm{for} \ \mathrm{any} \ h_{0} \in H_{0}\}, $

for a nonzero linear functional $ \alpha $ on $ H_{0} $ such that $ L_{\alpha}\neq 0 $.

We introduce the concept of a split regular BiHom-Lie color algebra in an analogous way.

Definition 2.8  We denote by $ H = \oplus_{g\in \Gamma}H_{g} $ a maximal abelian $ ( $graded$ ) $ subalgebra, of a regular BiHom-Lie color algebra $ L $. For a linear functional $ \alpha:H_{0}\rightarrow \mathbb{K}, $ we define the root space of $ L $ $ ( $with respect to $ H $$ ) $ associated to $ \alpha $ as the subspace

$ L_{\alpha} = \{v_{\alpha}\in L: [h_{0}, \phi(v_{\alpha})] = \alpha(h_{0})\phi\psi (v_{\alpha}), \ \mathrm{for }\ \mathrm{any} \ h_{0} \in H_{0}\}. $

The elements $ \alpha:H_{0}\rightarrow \mathbb{K} $ satisfying $ L_{\alpha}\neq 0 $ are called roots of $ L $ with respect to $ H $. We denote $ \Lambda: = \{\alpha \in (H_{0})^{\ast} \setminus \{0\}: L_{\alpha}\neq 0\} $. We say that $ L $ is a split regular BiHom-Lie color algebra, with respect to $ H $, if

$ L = H\oplus (\oplus_{\alpha \in \Lambda }L_{\alpha}). $

We also say that $ \Lambda $ is the root system of $ L $.

Noting that when $ \phi = \psi = \mathrm{Id} $, the split Lie color algebras become examples of split regular BiHom-Lie color algebras and when $ \phi = \psi $, the split regular Hom-Lie color algebras become examples of split regular BiHom-Lie color algebras. Hence, the present paper extends the results in [19, 22]. Let us see another example.

Example 2.9 Let $ (L = H\oplus (\oplus_{\alpha \in \Delta }L_{\alpha}), [\cdot, \cdot]) $ be a split Lie color algebra, $ \phi, \psi: L\rightarrow L $ two automorphisms such that $ \phi(H) = \psi(H) = H $ and $ \phi \circ \psi = \psi \circ \phi $. By the Example 2.6, we know that $ (L, [\cdot, \cdot]^{'}, \phi, \psi) $, where $ [x, y]^{'}: = [\phi(x), \psi(y)] $ for any element $ x, y \in L $, is a regular BiHom-Lie color algebra. Then it is straightforward to verify that the direct sum

$ L = H\oplus (\oplus_{\alpha \in \Delta }L_{\alpha\psi^{-1}}) $

makes of the regular BiHom-Lie color algebra $ (L, [\cdot, \cdot]^{'}, \phi, \psi) $ a split regular BiHom-Lie color algebra, being the root system $ \Lambda = \{\alpha\psi^{-1}: \alpha \in \Delta\} $.

From now on $ L = H\oplus(\oplus_{\alpha \in \Lambda}L_{\alpha}) $ denotes a split regular BiHom-Lie color algebras. Also, and for an easier notation, the mappings $ \phi|_{H}, \psi|_{H}, \phi|_{H}^{-1}, \psi|_{H}^{-1}: H\rightarrow H $ will be denoted by $ \phi $, $ \psi $, $ \phi^{-1} $, $ \psi^{-1} $ respectively.

It is clear that the root space associated to the zero root $ L_{0} $ satisfies $ H \subset L_{0} $. Conversely, given any $ v_{0} \in L_{0} $ we can write

$ v_{0} = h\oplus(\oplus_{i = 1}^{n}v_{\alpha_{i}}), $

where $ h \in H $ and $ v_{\alpha_{i}} \in L_{\alpha_{i}} $ for $ i = 1, \cdots, n, $ with $ \alpha_{i} \neq \alpha_{j} $ if $ i \neq j $. Hence

$ 0 = [h_{0}, h \oplus(\oplus_{i = 1}^{n}v_{\alpha_{i}}) ] = \oplus_{i = 1}^{n}\alpha_{i}(h_{0}) \psi(v_{\alpha_{i}}), $

for any $ h_{0}\in H_{0} $. So taking into account the direct character of the sum and that $ \alpha_{i}\neq 0 $ gives us $ v_{\alpha_{i}} = 0 $ for $ i = 1, \cdots, n $. So $ v_{0} = h \in H $. Consequently,

$ \begin{equation} H = L_{0}. \end{equation} $

(2.1)

Lemma 2.10 Let $ L = \oplus_{g \in \Gamma}L_{g} $ be a split BiHom-Lie color algebra with corresponding root space decomposition $ L = H\oplus (\oplus_{\alpha \in \Lambda }L_{\alpha}). $ If we denote by $ L_{\alpha, g} = L_{\alpha}\cap L_{g} $, then the following assertions hold.

$ \rm(1) $ $ L_{\alpha} = \oplus_{g \in \Gamma}L_{\alpha, g} $ for any $ \alpha \in \Lambda\cup \{0\} $.

$ \rm(2) $ $ H_{g} = L_{0, g}. $ In particular $ H_{0} = L_{0, 0} $.

$ \rm(3) $ $ L_{0} $ is a split BiHom-Lie algebra, respect to $ H_{0} $, with root space decomposition $ L_{0} = H_{0}\oplus (\oplus_{\alpha \in \Lambda }L_{\alpha, 0}). $

Proof  $ \rm(1) $ By the $ \Gamma $-grading of $ L $ we may express any $ v_{\alpha}\in L_{\alpha} $, $ \alpha \in \Lambda\cup \{0\} $, in the form $ v_{\alpha} = v_{\alpha, g_{1}}+\cdots+v_{\alpha, g_{n}} $ with $ v_{\alpha, g_{i}} \in L_{g_{i}} $ for distinct $ g_{1}, \cdots, g_{n} \in \Gamma $. If $ h_{0}\in H_{0} $ then $ [h_{0}, \phi(v_{\alpha, g_{i}})] = \alpha(h_{0})\phi\psi(v_{\alpha, g_{i}}) $ for $ i = 1, \cdots, n. $ Hence $ L_{\alpha} = \oplus_{g \in \Gamma}(L_{\alpha}\cap L_{g}) $ and we can write $ L_{\alpha} = \oplus_{g \in \Gamma}L_{\alpha, g} $ for any $ \alpha \in \Lambda\cup \{0\} $.

$ \rm(2) $ Consequence of (2.1) and item 1.

$ \rm(3) $ We also have $ L_{g} = H_{g}\oplus(\oplus_{\alpha \in \Lambda}L_{\alpha, g}) $ for any $ g\in \Gamma $. By considering $ g = 0 $ we get $ L_{0} = H_{0}\oplus(\oplus_{\alpha \in \Lambda}L_{\alpha, 0}). $ Hence, the direct character of the sum and the fact that $ \alpha \neq 0 $ for any $ \alpha \in \Lambda $ gives us that $ H_{0} $ is a maximal abelian subalgebra of the BiHom-Lie algebra $ L_{0} $. Hence $ L_{0} $ is a split BiHom-Lie algebra respect to $ H_{0} $.

Lemma 2.11 For any $ \alpha \in \Lambda \cup \{0\} $, the following assertions hold.

$ \rm(1) $ $ \phi(L_{\alpha}) = L_{\alpha \phi^{-1}} $ and $ \phi^{-1}(L_{\alpha}) = L_{\alpha \phi} $.

$ \rm(2) $ $ \psi(L_{\alpha}) = L_{\alpha \psi^{-1}} $ and $ \psi^{-1}(L_{\alpha}) = L_{\alpha \psi} $.

Proof  $ \rm(1) $ For any $ h_{0} \in H_{0} $ and $ v_{\alpha} \in L_{\alpha} $, since

$ \begin{equation} [h_{0}, \phi(v_{\alpha})] = \alpha(h_{0})\phi\psi(v_{\alpha}), \end{equation} $

(2.2)

we have that by writing $ h_{0}^{'} = \phi(h_{0}), $ then

$ \begin{align*} [h_{0}^{'}, \phi^{2}(v_{\alpha})]& = \phi([h_{0}, \phi(v_{\alpha})]) = \alpha(h_{0})\phi^{2}\psi (v_{\alpha})\\ & = \alpha\phi^{-1}(h_{0}^{'})\phi^{2}\psi (v_{\alpha}) = \alpha\phi^{-1}(h_{0}^{'})\phi\psi(\phi (v_{\alpha})). \end{align*} $

Therefore we get $ \phi(v_{\alpha}) \in L_{\alpha\phi^{-1}} $ and so

$ \begin{equation} \phi(L_{\alpha}) \subset L_{\alpha \phi^{-1}}. \end{equation} $

(2.3)

Now, let us show $ L_{\alpha \phi^{-1}} \subset \phi(L_{\alpha}). $ Indeed, for any $ h_{0} \in H_{0} $ and $ v_{\alpha} \in L_{\alpha} $, $ ( $2.2$ ) $ shows $ [\phi^{-1}(h_{0}), v_{\alpha}] = \alpha(h_{0})\psi(v_{\alpha}) $. From here, we get $ [\phi(h_{0}), v_{\alpha}] = \alpha\phi^{2}(h_{0})\psi(v_{\alpha}) $ and

$ \begin{align} \phi^{-1}(L_{\alpha})\subset L_{\alpha\phi}. \end{align} $

(2.4)

Hence, since for any $ x\in L_{\alpha\phi^{-1}} $ we can write $ x = \phi(\phi^{-1}(x)) $ and by $ ( $2.4$ ) $ we have $ \phi^{-1}(x)\in L_{\alpha} $ and $ L_{\alpha\phi^{-1}}\subset \phi(L_{\alpha}) $. This fact together with $ ( $2.3$ ) $ show $ \phi(L_{\alpha}) = L_{\alpha \phi^{-1}} $.

To show $ \phi^{-1}(L_{\alpha}) = L_{\alpha \phi}, $ the fact $ \phi^{-1}(L_{\alpha})\subset L_{\alpha \phi} $ is $ ( $2.4$ ) $, while the fact $ L_{\alpha \phi} \subset \phi^{-1}(L_{\alpha}) $ is consequence of writing any element $ x\in L_{\alpha \phi} $ of the form $ x = \phi^{-1}(\phi(x)) $ and applying $ ( $2.3$ ) $.

$ \rm(2) $ To verify

$ \begin{align} \psi(L_{\alpha})\subset L_{\alpha\psi^{-1}}, \end{align} $

(2.5)

we observe that (2.2) gives us $ [\psi(h_{0}), \psi\phi(v_{\alpha})] = \alpha(h_{0})\psi\phi\psi(v_{\alpha}) $, and so $ [\psi(h_{0}), \phi\psi(v_{\alpha})] = \alpha\psi^{-1}(\psi(h_{0}))\phi\psi(\psi(v_{\alpha})) $. Since (2.2) and the identity $ \psi^{-1}\phi = \phi\psi^{-1} $ also gives us

$ \begin{align} \psi^{-1}(L_{\alpha})\subset L_{\alpha\psi}, \end{align} $

(2.6)

we conclude as above that $ \psi(L_{\alpha}) = L_{\alpha\psi^{-1}} $. We can argue similarly with (2.5) and (2.6) to get $ \psi^{-1}(L_{\alpha}) = L_{\alpha \psi} $.

Lemma 2.12 For any $ \alpha $, $ \beta \in \Lambda \cup \{0\} $, we have $ [L_{\alpha}, L_{\beta}] \subset L_{\alpha\phi^{-1}+\beta\psi^{-1}} $.

Proof  For each $ h_{0} \in H_{0} $, $ v_{\alpha} \in L_{\alpha} $ and $ v_{\beta} \in L_{\beta} $, we can write

$ [h_{0}, \phi([v_{\alpha}, v_{\beta}])] = [\psi^{2}\psi^{-2}(h_{0}), \phi([v_{\alpha}, v_{\beta}])]. $

So, by denoting $ h_{0}^{'} = \psi^{-2}(h_{0}) $, we can apply BiHom-Jacobi identity and BiHom-skew-symmetry to get

$ \begin{align*} &\quad[\psi^{2}(h_{0}^{'}), \phi([v_{\alpha}, v_{\beta}])]\\ & = [\psi^{2}(h_{0}^{'}), [\psi\psi^{-1}\phi(v_{\alpha}), \phi(v_{\beta})]]\\ & = -\varepsilon(\bar{h_{0}^{'}}, \bar{\alpha}+\bar{\beta})[\psi\phi(v_{\alpha}), [\psi(v_{\beta}), \phi(h_{0}^{'})]]-\varepsilon(\bar{\alpha}+\bar{h_{0}^{'}}, \bar{\beta})[\psi^{2}(v_{\beta}), [\psi(h_{0}^{'}), \phi\psi^{-1}\phi(v_{\alpha})]]\\ & = -\varepsilon(\bar{h_{0}^{'}}, \bar{\alpha}+\bar{\beta})(-\varepsilon(\bar{\beta}, \bar{h_{0}^{'}}))[\psi\phi(v_{\alpha}), [\psi(h_{0}^{'}), \phi(v_{\beta})]]-\varepsilon(\bar{\alpha}+\bar{h_{0}^{'}}, \bar{\beta})[\psi^{2}(v_{\beta}), [\psi(h_{0}^{'}), \phi\psi^{-1}\phi(v_{\alpha})]]\\ & = \varepsilon(\bar{h_{0}^{'}}, \bar{\alpha})[\psi\phi(v_{\alpha}), [\psi(h_{0}^{'}), \phi(v_{\beta})]]-\varepsilon(\bar{\alpha}+\bar{h_{0}^{'}}, \bar{\beta})[\psi^{2}(v_{\beta}), \phi[\phi^{-1}\psi(h_{0}^{'}), \psi^{-1}\phi(v_{\alpha})]]\\ & = \varepsilon(\bar{h_{0}^{'}}, \bar{\alpha})[\psi\phi(v_{\alpha}), [\psi(h_{0}^{'}), \phi(v_{\beta})]] -\varepsilon(\bar{\alpha}+\bar{h_{0}^{'}}, \bar{\beta})(-\varepsilon(\bar{\beta}, \bar{\alpha}+\bar{h_{0}^{'}}))[\psi[\phi^{-1}\psi(h_{0}^{'}), \psi^{-1}\phi(v_{\alpha})], \phi\psi(v_{\beta})] \\ & = \varepsilon(\bar{h_{0}^{'}}, \bar{\alpha})[\psi\phi(v_{\alpha}), [\psi(h_{0}^{'}), \phi(v_{\beta})]]+[[\psi^{2}\phi^{-1}(h_{0}^{'}), \phi(v_{\alpha})], \phi\psi(v_{\beta})] \\ & = \varepsilon(\bar{h_{0}^{'}}, \bar{\alpha})\beta\psi(h_{0}^{'})[\psi\phi(v_{\alpha}), \phi \psi(v_{\beta})]+\alpha(\psi^{2}\phi^{-1}(h_{0}^{'}))[\phi\psi(v_{\alpha}), \phi\psi(v_{\beta})] \\ & = \varepsilon(\bar{h_{0}^{'}}, \bar{\alpha})(\beta\psi+\alpha\psi^{2}\phi^{-1})(h_{0}^{'})[\psi\phi(v_{\alpha}), \phi \psi(v_{\beta})] \\ & = \varepsilon(\bar{h_{0}^{'}}, \bar{\alpha})(\beta\psi+\alpha\psi^{2}\phi^{-1})(h_{0}^{'})[\phi\psi(v_{\alpha}), \phi \psi(v_{\beta})] \\ & = \varepsilon(\bar{h_{0}^{'}}, \bar{\alpha})(\beta\psi+\alpha\psi^{2}\phi^{-1})(h_{0}^{'})\phi\psi([v_{\alpha}, v_{\beta}])\\ & = (\beta\psi+\alpha\psi^{2}\phi^{-1})(h_{0}^{'})\phi\psi([v_{\alpha}, v_{\beta}]). \end{align*} $

Taking into account $ h_{0}^{'} = \psi^{-2}(h_{0}) $ we have shown that

$ [h_{0}, \phi([v_{\alpha}, v_{\beta}])] = (\beta\psi^{-1}+\alpha\phi^{-1})(h_{0})\phi\psi([v_{\alpha}, v_{\beta}]). $

From here, $ [L_{\alpha}, L_{\beta}] \subset L_{\alpha\phi^{-1}+\beta\psi^{-1}} $.

From Lemma 2.12 we can assert that

$ [L_{\alpha, g_{1}}, L_{\beta, g_{2}}] \subset L_{\alpha\phi^{-1}+\beta\psi^{-1}, g_{1}+g_{2}} $

for any $ g_{1}, g_{2}\in \Gamma $.

Lemma 2.13 If $ \alpha \in \Lambda $, then $ \alpha\phi^{-z_{1}}\psi^{-z_{2}}\in \Lambda $ for any $ z_{1}, z_{2}\in \mathbb{Z} $.

Proof  This is a consequence of Lemma 2.11 (1) and (2).

Definition 2.14 A root system $ \Lambda $ of a split BiHom-Lie color algebra is called symmetric if it satisfies that $ \alpha \in \Lambda $ implies $ -\alpha \in \Lambda $.

In the following, let $ L $ be a split regular BiHom-Lie color algebra with a symmetric root system $ \Lambda $ and $ L = H \oplus(\oplus _{\alpha \in \Lambda}L_{\alpha}) $ the corresponding root decomposition. We begin by developing the techniques of connections of roots in this section.

Definition 3.1 Let $ \alpha $ and $ \beta $ be two nonzero roots. We shall say that $ \alpha $ is connected to $ \beta $ if there exists $ \alpha_{1}, \cdots, \alpha_{k}\in \Lambda $ such that

If $ k = 1 $, then $ \alpha_{1}\in \{\alpha\phi^{-n}\psi^{-r}:n, r\in \mathbb{N}\}\cap \{\pm \beta\phi^{-m}\psi^{-s}: m, s\in \mathbb{N} \} $.

If $ k\geq 2 $, then

$ \rm(1) $ $ \alpha_{1}\in \{\alpha\phi^{-n}\psi^{-r}:n, r\in \mathbb{N}\}. $

$ \rm(2) $ $ \alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1} \in \Lambda $,

$\quad \alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-1}\psi^{-1}+\alpha_{3}\psi^{-1} \in \Lambda $,

$\quad\quad \vdots $

$\quad \alpha_{1}\phi^{-i}+\alpha_{2}\phi^{-i+1}\psi^{-1}+\alpha_{3}\phi^{-i+2}\psi^{-1}+\cdots+\alpha_{i}\phi^{-1}\psi^{-1}+\alpha_{i+1}\psi^{-1}\in \Lambda $,

$\quad\quad \vdots $

$\quad \alpha_{1}\phi^{-k+2}+\alpha_{2}\phi^{-k+3}\psi^{-1}+\alpha_{3}\phi^{-k+4}\psi^{-1}+\cdots+\alpha_{k-2}\phi^{-1}\psi^{-1}+\alpha_{k-1}\psi^{-1}\in \Lambda $.

$ \rm(3) $ $ \alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+2}\psi^{-1}+\alpha_{3}\phi^{-k+3}\psi^{-1}+\cdots+\alpha_{i}\phi^{-k+i}\psi^{-1}+\cdots+\alpha_{k-1}\phi^{-1}\psi^{-1}+\alpha_{k}\psi^{-1}\in \{\pm \beta\phi^{-m}\psi^{-s}: m, s\in \mathbb{N} \} $.

We shall also say that $ \{\alpha_{1}, \cdots, \alpha_{k}\} $ is a connection from $ \alpha $ to $ \beta $.

Our next goal is to show that the connection is an equivalence relation on $ \Lambda $.

Proposition 3.2 The relation $ \sim $ in $ \Lambda $, defined by $ \alpha \sim \beta $ if and only if $ \alpha $ is connected to $ \beta $, is an equivalence relation.

Proof  This can be proved completely analogously to [14, Corollary 2.1].

For any $ \alpha \in \Lambda $, we denote by

$ \Lambda_{\alpha}: = \{\beta \in \Lambda : \beta\sim \alpha\}. $

Clearly if $ \beta \in \Lambda_{\alpha} $ then $ -\beta \in \Lambda_{\alpha} $ and, by Proposition, if $ \gamma \not \in \Lambda_{\alpha} $ then $ \Lambda_{\alpha}\cap \Lambda_{\gamma} = \emptyset $.

Our next goal is to associate an adequate ideal $ L_{\Lambda_{\alpha}} $ of $ L $ to any $ \Lambda_{\alpha} $. For $ \Lambda_{\alpha} $, $ \alpha \in \Lambda $, we define $ H_{\Lambda_{\alpha}}: = \mathrm{span_{\mathbb{K}}}\{[L_{\beta\psi^{-1}}, L_{-\beta\phi^{-1}}]: \beta \in \Lambda_{\alpha}\}, $ and $ V_{\Lambda_{\alpha}}: = \oplus_{\beta \in \Lambda_{\alpha}}L_{\beta}. $ We denote by $ L_{\Lambda_{\alpha}} $ the following graded subspace of $ L $, $ L_{\Lambda_{\alpha}}: = H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}}. $

Proposition 3.3  For any $ \alpha \in \Lambda $, the linear subspace $ L_{\Lambda_{\alpha}} $ is a subalgebra of $ L $.

Proof  First we have to check that $ L_{\Lambda_{\alpha}} $ satisfies $ [L_{\Lambda_{\alpha}}, L_{\Lambda_{\alpha}}]\subset L_{\Lambda_{\alpha}}. $ Taking into account $ H = L_{0} $, then $ [H_{\Lambda_{\alpha}}, H_{\Lambda_{\alpha}}] = 0 $ and

$ \begin{equation} [ L_{\Lambda_{\alpha}}, L_{\Lambda_{\alpha}}] = [H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}}, H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}}] \subset [H_{\Lambda_{\alpha}}, V_{\Lambda_{\alpha}}] + [V_{\Lambda_{\alpha}}, H_{\Lambda_{\alpha}}] +\Sigma_{\beta, \gamma \in \Lambda_{\alpha}}[L_{\beta}, L_{\gamma}]. \end{equation} $

(3.1)

Let us consider the first summand in (3.1). Given $ \beta \in \Lambda_{\alpha} $, we have $ [H_{\Lambda_{\alpha}}, L_{\beta}]\subset [L_{0}, L_{\beta}]\subset L_{\beta\psi^{-1} } $, being $ \beta\psi^{-1} \in \Lambda_{\alpha} $ by Lemma 2.13. Hence,

$ \begin{equation} [H_{\Lambda_{\alpha}}, V_{\Lambda_{\alpha}}]\subset V_{\Lambda_{\alpha}}. \end{equation} $

(3.2)

Similarly, we can also get

$ \begin{equation} [V_{\Lambda_{\alpha}}, H_{\Lambda_{\alpha}}]\subset V_{\Lambda_{\alpha}}. \end{equation} $

(3.3)

We consider now the third summand $ \Sigma_{\beta, \gamma\in \Lambda_{\alpha}}[L_{\beta}, L_{\gamma}] $. Given $ \beta, \gamma \in \Lambda_{\alpha} $ such that $ [L_{\beta}, L_{\gamma}]\neq 0 $, if $ \beta\phi^{-1}+\gamma\psi^{-1} = 0 $, then clearly $ [L_{\beta}, L_{\gamma}]\subset H_{\Lambda_{\alpha}}. $ Supposing that $ \beta\phi^{-1}+\gamma\psi^{-1}\neq 0 $, since $ [L_{\beta}, L_{\gamma}]\neq 0 $ together with Lemma 2.12 ensures that $ \beta\phi^{-1}+\gamma\psi^{-1} \in \Lambda $, we have that $ \{\beta, \gamma\} $ is a connection from $ \beta $ to $ \beta\phi^{-1}+\gamma\psi^{-1} $. The transitivity of $ \sim $ gives now that $ \beta\phi^{-1}+\gamma\psi^{-1} \in \Lambda_{\alpha} $ and so

$ \begin{equation} [L_{\beta}, L_{\gamma}]\subset L_{\beta\phi^{-1}+\gamma\psi^{-1}}\subset V_{ \Lambda_{\alpha}}. \end{equation} $

(3.4)

From (3.1)-(3.4), we conclude that $ [L_{\Lambda_{\alpha}}, L_{\Lambda_{\alpha}}]\subset L_{\Lambda_{\alpha}}. $

Secondly, we have to verify that $ \phi(L_{\Lambda_{\alpha}}) = L_{\Lambda_{\alpha}} $ and $ \psi(L_{\Lambda_{\alpha}}) = L_{\Lambda_{\alpha}} $. But this is a direct consequence of Lemma 2.11.

Proposition 3.4  If $ \gamma \not \in \Lambda_{\alpha} $, then $ [L_{\Lambda_{\alpha}}, L_{\Lambda_{\gamma}}] = 0 $.

Proof  We have

$ \begin{equation} [ L_{\Lambda_{\alpha}}, L_{\Lambda_{\gamma}}] = [H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}}, H_{\Lambda_{\gamma}}\oplus V_{\Lambda_{\gamma}}] \subset [H_{\Lambda_{\alpha}}, V_{\Lambda_{\gamma}}] + [V_{\Lambda_{\alpha}}, H_{\Lambda_{\gamma}}] +[V_{\Lambda_{\alpha}}, V_{\Lambda_{\gamma}}]. \end{equation} $

(3.5)

We consider the above third summand $ [V_{\Lambda_{\alpha}}, V_{\Lambda_{\gamma}}] $ and suppose that there exist $ \beta \in \Lambda_{\alpha} $ and $ \eta \in \Lambda_{\gamma} $ such that $ [L_{\beta}, L_{\eta}]\neq 0 $. As necessarily $ \beta\phi^{-1} \neq -\eta\psi^{-1} $, then $ \beta\phi^{-1}+\eta\psi^{-1} \in \Lambda $. So $ \{\beta, \eta, -\beta\phi^{-2}\psi\} $ is a connection between $ \beta $ and $ \eta $. By the transitivity of the connection relation we have $ \gamma \in \Lambda_{\alpha} $, a contradiction. Hence $ [L_{\beta}, L_{\eta}] = 0 $ and so

$ \begin{equation} [V_{\Lambda_{\alpha}}, V_{\Lambda_{\gamma}}] = 0. \end{equation} $

(3.6)

We consider now the first summand $ [H_{\Lambda_{\alpha}}, V_{\Lambda_{\gamma}}] $ in (3.5) and suppose there exist $ \beta \in \Lambda_{\alpha} $ and $ \eta \in \Lambda_{\gamma} $ such that

$ [[L_{\beta\psi^{-1}}, L_{-\beta\phi^{-1}}], \phi^{2}(L_{\eta})]\neq 0. $

By BiHom-skew-symmetry, $ [\psi^{2}(L_{\eta}), [L_{-\beta\psi^{-1}}, L_{\beta\phi^{-1}}]]\neq 0 $. Hence, there exist $ i, j, k\in \Gamma $ such that $ [\psi^{2}(L_{\eta, i}), [\psi (L_{-\beta, j}), \phi (L_{\beta, k})]]\neq 0 $. By BiHom-Jacobi identity, we get either $ [\psi(L_{\beta, k}), \phi(L_{\eta, i})]\neq 0 $ or $ [\psi(L_{\eta, i}), \phi(L_{-\beta, j})]\neq 0 $. From here $ [V_{\Lambda_{\alpha}}, V_{\Lambda_{\gamma}}]\neq 0 $ in any case, which contradicts (3.6). Hence $ [H_{\Lambda_{\alpha}}, V_{\Lambda_{\gamma}}] = 0. $ Finally, we note that the same above argument shows, $ [V_{\Lambda_{\alpha}}, H_{\Lambda_{\gamma}}] = 0. $ By (3.5), we conclude $ [L_{\Lambda_{\alpha}}, L_{\Lambda_{\gamma}}] = 0 $.

Theorem 3.5 The following assertions hold.

$ \rm(1) $ For any $ \alpha \in \Lambda $, the subalgebra $ L_{\Lambda_{\alpha}} = H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}} $ of $ L $ associated to $ \Lambda_{\alpha} $ is an ideal of $ L $.

$ \rm(2) $ If $ L $ is simple, then there exists a connection from $ \alpha $ to $ \beta $ for any $ \alpha, \beta \in \Lambda $ and $ H = \sum_{\alpha \in \Lambda}[L_{\alpha\psi^{-1}}, L_{-\alpha\phi^{-1}}] $.

Proof  $ \rm(1) $ Since $ [L_{\Lambda_{\alpha}}, H] = [L_{\Lambda_{\alpha}}, L_{0}]\subset V_{\Lambda_{\alpha}} $, taking into account Propositions 3.3 and 3.4, we have

$ [L_{\Lambda_{\alpha}}, L] = [L_{\Lambda_{\alpha}}, H\oplus(\oplus_{\beta \in \Lambda_{\alpha}}L_{\beta})\oplus(\oplus_{\gamma \not \in \Lambda_{\alpha}}L_{\gamma})]\subset L_{\Lambda_{\alpha}}. $

In a similar way we get $ [L, L_{\Lambda_{\alpha}}]\subset L_{\Lambda_{\alpha}} $. Finally, by Lemma 2.11, we also have $ \phi(L_{\Lambda_{\alpha}}) = L_{\Lambda_{\alpha}} $ and $ \psi(L_{\Lambda_{\alpha}}) = L_{\Lambda_{\alpha}} $. So we conclude that $ L_{\Lambda_{\alpha}} $ is an ideal of $ L $.

$ \rm(2) $ The simplicity of $ L $ implies $ L_{\Lambda_{\alpha}} = L $. From here, it is clear that $ \Lambda_{\alpha} = \Lambda $ and $ H = \sum_{\alpha \in \Lambda}[L_{\alpha\psi^{-1}}, L_{-\alpha\phi^{-1}}] $.

Theorem 3.6 For a vector space complement $ U $ of span$ _{\mathbb{K}}\{{[L_{\alpha\psi^{-1}}, L_{-\alpha\phi^{-1}}]: \alpha \in \Lambda}\} $ in H, we have

$ L = U + \sum\limits_{[\alpha] \in \Lambda/\sim}I_{[\alpha]}, $

where any $ I_{[\alpha]} $ is one of the ideals $ L_{\Lambda_{\alpha}} $ of $ L $ described in Theorem 3.5(1), satisfying $ [I_{[\alpha]}, I_{[\beta]}] = 0, $ whenever $ [\alpha] \neq [\beta]. $

Proof  By Proposition 3.2, we can consider the quotient set $ \Lambda/\sim : = \{[\alpha]: \alpha \in \Lambda\} $. Let us denote by $ I_{[\alpha]}: = L_{\Lambda_{\alpha}} $. We obtain that $ I_{[\alpha]} $ is well defined and by Theorem 3.5(1), an ideal of $ L $. Therefore

$ L = U + \sum\limits_{[\alpha] \in \Lambda/\sim}I_{[\alpha]}. $

By applying Proposition 3.4 we also obtain $ [I_{[\alpha]}, I_{[\beta]}] = 0 $ if $ [\alpha] \neq [\beta]. $

Let us denote by $ \mathrm{Z}(L): = \{x \in L: [x, L] +[L, x ] = 0\} $ the center of $ L $.

Corollary 3.7 If $ \mathrm{Z}(L) = 0 $ and $ H = \sum_{\alpha \in \Lambda}[L_{\alpha\psi^{-1}}, L_{-\alpha\phi^{-1}}] $, then $ L $ is the direct sum of the ideals given in Theorem 3.5,

$ L = \oplus_{[\alpha] \in \Lambda/\sim}I_{[\alpha]}. $

Furthermore $ [I_{[\alpha]}, I_{[\beta]}] = 0, $ whenever $ [\alpha] \neq [\beta]. $

Proof  Since $ H = \sum_{\alpha \in \Lambda}[L_{\alpha\psi^{-1}}, L_{-\alpha\phi^{-1}}] $, we get $ L = \oplus_{[\alpha] \in \Lambda/\sim}I_{[\alpha]}. $ We show the direct character of the sum. Given $ x\in I_{[\alpha]}\cap \sum_{[\beta]\in \Lambda/\sim \atop{ [\beta]\neq [\alpha]} }I_{[\beta]} $, by using again the equation $ [I_{[\alpha]}, I_{[\beta]}] = 0 $, for $ [\alpha] \neq [\beta] $, we obtain

$ [x, I_{[\alpha]}]+[x, \sum\limits_{[\beta]\in \Lambda/\sim \atop{[\beta]\neq [\alpha]} }I_{[\beta]}] = 0, $

$ [I_{[\alpha]}, x]+[\sum\limits_{[\beta]\in \Lambda/\sim \atop{[\beta]\neq [\alpha]} }I_{[\beta]}, x] = 0. $

It implies $ [x, L] +[L, x ] = 0 $, that is, $ x\in \mathrm{Z}(L) = 0 $. Thus $ x = 0 $, as desired.

In this section, we study the sufficient conditions for the decomposition of $ L $ into direct sums of simple ideals. Under certain conditions we give an affirmative answer.

Lemma 4.1  Let $ L = H\oplus(\oplus_{\alpha \in \Lambda}L_{\alpha}) $ be a split regular BiHom-Lie color algebra. If $ I $ is an ideal of $ L $, then $ I = (I\cap H)\oplus(\oplus_{\alpha \in \Lambda}(I\cap L_{\alpha})). $

Proof  We can see $ L = H\oplus(\oplus_{\alpha \in \Lambda}L_{\alpha}) $ as a weight module with respect to the split BiHom-Lie algebra $ L_{0} $, with maximal abelian subalgebra $ H_{0} $, in the natural way. The character of ideal of $ I $ gives us that $ I $ is a submodule of $ L $. It is well-known that a submodule of a weight module is again a weight module. From here, $ I $ is a weight module with respect to $ L_{0} $ $ ( $and $ H_{0} $$ ) $ and so $ I = (I\cap H)\oplus(\oplus_{\alpha \in \Lambda}(I\cap L_{\alpha})). $

Taking into account the above lemma, we observe that the grading of $ I $ and Lemma 2.10(1) let us write

$ \begin{equation} I = \oplus_{g \in \Gamma}I_{g} = \oplus_{g \in \Gamma}\big((I_{g}\cap H_{g})\oplus (\oplus_{\alpha \in \Lambda}(I_{g}\cap L_{\alpha, g})\big). \end{equation} $

(4.1)

Lemma 4.2 Let $ L $ be a split regular BiHom-Lie color algebra with $ \mathrm{Z}(L) = 0 $ and $ I $ an ideal of $ L $. If $ I\subset H $, then $ I = \{0\} $.

Proof  We suppose that there exists a nonzero ideal $ I $ of $ L $ such that $ I \subset H $. We get $ [I, H] \subset [H, H] = 0 $ and $ [I, \oplus_{\alpha \in \Lambda}L_{\alpha}]\subset I\subset H $. Then taking into account $ H = L_{0} $, we have $ [I, \oplus_{\alpha \in \Lambda}L_{\alpha}]\subset H \cap (\oplus_{\alpha \in \Lambda}L_{\alpha}) = 0 $ and $ [\oplus_{\alpha \in \Lambda}L_{\alpha} , I ]\subset (\oplus_{\alpha \in \Lambda}L_{\alpha})\cap H = 0. $ From here $ I\subset \mathrm{Z}(L) = 0 $, which is a contradiction.

Let us introduce the concepts of root-multiplicativity and maximal length in the framework of split Hom-Lie color algebras. For each $ g \in \Gamma $, we denote $ \Lambda_{g}: = \{\alpha \in \Lambda: L_{\alpha, g}\neq 0\}. $

Definition 4.3  A split regular BiHom-Lie color algebra $ L $ is root-multiplicative if given $ \alpha \in \Lambda_{g_{i}} $ and $ \beta \in \Lambda_{g_{j}} $, with $ g_{i}, g_{j}\in \Gamma $, such that $ \alpha+\beta \in \Lambda $, then $ [L_{\alpha, g_{i}}, L_{\beta, g_{j}}]\neq 0 $.

Definition 4.4  A split regular BiHom-Lie color algebra $ L $ is of maximal length if for any $ \alpha \in \Lambda_{g}, g\in \Gamma $, we have $ \mathrm{dim}L_{\kappa\alpha, \kappa g} = 1 $ for $ \kappa \in \{\pm1\} $.

If $ L $ is of maximal length, according to (4.1) we assert that given any nonzero ideal $ I $ of $ L $ then

$ \begin{equation} I = \oplus_{g \in \Gamma}\big((I_{g}\cap H_{g})\oplus (\oplus_{\alpha \in \Lambda_{g}^{I}} L_{\alpha, g})\big). \end{equation} $

(4.2)

where $ \Lambda_{g}^{I}: = \{\alpha \in \Lambda: I_{g}\cap L_{\alpha, g}\neq 0\} $ for each $ g \in \Gamma $.

Theorem 4.5  Let $ L $ be a split regular BiHom-Lie color algebra of maximal length, root multiplicative and $ \mathrm{Z}(L) = 0 $. Then $ L $ is simple if and only if it has all of its nonzero roots connected and $ H = \sum_{\alpha \in \Lambda}[L_{\alpha\psi^{-1}}, L_{-\alpha\phi^{-1}}] $.

Proof  The first implication is Theorem 3.5(2). To prove the converse, we consider $ I $ a nonzero ideal of $ L $. By Lemma 4.2 and (4.2) we can write $ I = \oplus_{g \in \Gamma}\big((I_{g}\cap H_{g})\oplus (\oplus_{\alpha \in \Lambda_{g}^{I}} L_{\alpha, g})\big) $ with $ \Lambda_{g}^{I} \subset \Lambda_{g} $ for any $ g \in \Gamma $ and some $ \Lambda_{g}^{I} \neq \emptyset $. Hence, we may choose $ \alpha_{0} \in \Lambda_{g}^{I} $ such that

$ \begin{equation} 0\neq L_{\alpha_{0}, g}\subset I. \end{equation} $

(4.3)

Since $ \phi(I) = I $, $ \psi(I) = I $, and by Lemma 2.11, we can assert that $ \mbox{if} \ \alpha \in \Lambda_{I}, \ \mbox{then}\ \{\alpha\phi^{z_{1}}\psi^{z_{2}}:z_{1}, z_{2} \in \mathbb{Z}\}\subset \Lambda_{I}. $ In particular,

$ \begin{align*} \{L_{\alpha_{0}\phi^{z_{1}}\psi^{z_{2}}, g}:z_{1}, z_{2} \in \mathbb{Z}\}\subset I. \end{align*} $

Now, let us take any $ \beta \in \Lambda $ satisfying $ \beta \not \in \{\pm \alpha_{0}\phi^{z_{1}}\psi^{z_{2}}:z_{1}, z_{2} \in \mathbb{Z}\} $. Since $ \alpha_{0} $ and $ \beta $ are connected, we have a connection $ \{\alpha_{1}, \cdots, \alpha_{k}\} $, $ k\geq 2 $, from $ \alpha_{0} $ to $ \beta $ satisfying:

$\quad \alpha_{1} = a_{0}\phi^{-n}\psi^{-r} $, for some $ n, r\in \mathbb{N} $,

$\quad \alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1} \in \Lambda $,

$\quad \alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-1}\psi^{-1}+\alpha_{3}\psi^{-1} \in \Lambda $,

$ \qquad\qquad \vdots $

$\quad \alpha_{1}\phi^{-i}+\alpha_{2}\phi^{-i+1}\psi^{-1}+\alpha_{3}\phi^{-i+2}\psi^{-1}+\cdots+\alpha_{i}\phi^{-1}\psi^{-1}+\alpha_{i+1}\psi^{-1}\in \Lambda $,

$ \qquad\qquad \vdots $

$ \quad\alpha_{1}\phi^{-k+2}+\alpha_{2}\phi^{-k+3}\psi^{-1}+\alpha_{3}\phi^{-k+4}\psi^{-1}+\cdots+\alpha_{k-2}\phi^{-1}\psi^{-1}+\alpha_{k-1}\psi^{-1}\in \Lambda $,

$\quad \alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+2}\psi^{-1}+\alpha_{3}\phi^{-k+3}\psi^{-1}+\cdots+\alpha_{i}\phi^{-k+i}\psi^{-1}+\cdots+\alpha_{k-1}\phi^{-1}\psi^{-1}+\alpha_{k}\psi^{-1} = \epsilon \beta\phi^{-m}\psi^{-s} $ for some $ m, s\in \mathbb{N} $ and $ \epsilon \in \{\pm 1\} $.

Since $ \alpha_{2} \in \Lambda $, there exists $ g_{1} \in \Gamma $ such that $ L_{\alpha_{2}, g_{1}}\neq 0 $ and so $ \alpha_{2} \in \Lambda_{ g_{1}} $. From here, we have $ \alpha_{1}\in \Lambda_{g} $ and $ \alpha_{2}\in \Lambda_{g_{1}} $, such that $ \alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1}\in \Lambda_{g+g_{1}} $. The root-multiplicativity and maximal length of $ L $ show $ 0\neq [L_{\alpha_{1}, g}, L_{\alpha_{2}, g_{1}}] = L_{\alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1}, g+g_{1}} $. Since $ 0\neq L_{\alpha_{1}, g}\subset I $ as the consequence of (4.3) we get

$ 0\neq L_{\alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1}, g+g_{1}} \subset I. $

We can argue in a similar way from $ \alpha_{1}\phi^{-1}+\alpha_{2}\psi^{-1} $, $ \alpha_{3} $ and $ \alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-1}\psi^{-1} +\alpha_{3}\psi^{-1} $ to get

$ 0\neq L_{\alpha_{1}\phi^{-2}+\alpha_{2}\phi^{-1}\psi^{-1} +\alpha_{3}\psi^{-1}, g_{2}}\subset I $

for some $ g_{2} \in \Gamma $. Following this process with the connection $ \{\alpha_{1}, \cdots, \alpha_{k}\} $, we obtain that

$ 0\neq L_{\alpha_{1}\phi^{-k+1}+\alpha_{2}\phi^{-k+2}\psi^{-1}+\cdots+\alpha_{k}\psi^{-1}, g_{3}} \subset I $

and so either $ 0\neq L_{\beta\phi^{-m}\psi^{-s}, g_{3}}\subset I $ or $ 0\neq L_{-\beta\phi^{-m}\psi^{-s}, g_{3}}\subset I $ for some $ g_{3} \in \Gamma $. That is,

$ \begin{equation*} \label{czhangjian1234888} 0\neq L_{\epsilon\beta\phi^{-m}\psi^{-s}, g_{3}}\subset I \quad \mathrm{for} \quad \mathrm{some} \quad \epsilon \in\{\pm 1\}, \quad \mathrm{some} \quad g_{3}\in \Gamma \end{equation*} $

and for any $ \beta \in \Lambda $. By Lemma 2.11, we can get

$ \begin{equation} 0\neq L_{\epsilon\beta, g_{3}}\subset I \quad \mathrm{for} \quad \mathrm{some} \quad \epsilon \in\{\pm 1\}, \quad \mathrm{some} \quad g_{3}\in \Gamma. \end{equation} $

(4.4)

Taking into account $ H = \sum_{\beta \in \Lambda}[L_{\beta\psi^{-1}}, L_{-\beta\phi^{-1}}] $, the grading of $ L $ gives

$ H_{0} = \sum\limits_{\gamma \in \Lambda, g \in \Gamma}[\psi(L_{\gamma, g}), \phi(L_{-\gamma, -g})]. $

From here, there exists $ \gamma \in \Lambda $ and $ g_{4} \in \Gamma $ such that

$ \begin{equation} [[\psi(L_{\gamma, g_{4}}), \phi(L_{-\gamma, -g_{4}})], \psi^{2}\psi^{-2}\phi(L_{\epsilon\beta, g_{3}})]\neq 0. \end{equation} $

(4.5)

By the BiHom-Jacobi identity either

$ [\psi(L_{-\gamma, -g_{4}}), \phi(\psi^{-2}\phi(L_{\epsilon\beta, g_{3}}))]\neq 0 \quad \text{or} \quad[\psi(\psi^{-2}\phi(L_{\epsilon\beta, g_{3}})), \phi(L_{\gamma, g_{4}})]\neq 0 $

and so

$ L_{-\gamma\psi^{-1}\phi^{-1}+\epsilon\beta\psi\phi^{-2}, -g_{4}+g_{3}}\neq 0\quad \text{or} \quad L_{\gamma\psi^{-1}\phi^{-1}+\epsilon\beta\psi\phi^{-2}, g_{4}+g_{3}}\neq 0. $

That is

$ \begin{equation} 0\neq L_{\kappa\gamma\psi^{-1}\phi^{-1}+\epsilon\beta\psi\phi^{-2}, \kappa g_{4}+g_{3}}\subset I \end{equation} $

(4.6)

for some $ \kappa \in \{\pm 1\}. $ Since $ \epsilon\beta \in \Lambda_{g_{3}} $, by the maximal length of $ L $ we have $ -\epsilon\beta \in \Lambda_{-g_{3}} $. By (4.6) and the root-multiplicativity and maximal length of $ L $ we obtain

$ \begin{equation} 0\neq [ L_{\kappa\gamma\psi^{-1}\phi^{-1}+\epsilon\beta\psi\phi^{-2}, \kappa g_{4}+g_{3}}, L_{-\epsilon\beta\phi^{-3}\psi^{2}, -g_{3}}] = L_{\kappa\gamma\psi^{-1}\phi^{-2}, \kappa g_{4}}\subset I. \end{equation} $

(4.7)

By Lemma 2.11(1), we can get

$ \begin{equation} L_{\kappa\gamma, \kappa g_{4}}\subset I. \end{equation} $

(4.8)

Taking into account (4.7) and (4.5) we get $ \beta\phi^{-1}([\psi(L_{\gamma, g_{4}}), \phi(L_{-\gamma, -g_{4}})])\neq 0. $ For any $ g_{5} \in \Gamma $ such that $ L_{\epsilon\beta, g_{5}}\neq 0 $, we have

$ 0\neq[[L_{\gamma\psi^{-1}, g_{4}}, L_{-\gamma\phi^{-1}, -g_{4}}], \phi(L_{\epsilon\beta, g_{5}})] = L_{\epsilon\beta\phi^{-1}\psi^{-1}, g_{5}}\subset I $

and so $ L_{\epsilon\beta}\subset I $. That is, we can assert that

$ \begin{equation} L_{\epsilon\beta}\subset I \end{equation} $

(4.9)

for any $ \beta \in \Lambda $ and some $ \epsilon \in \{\pm 1\} $. Since $ H = \sum_{\beta \in \Lambda}[L_{\beta\psi^{-1}}, L_{-\beta\phi^{-1}}], $ we get

$ \begin{equation} H\subset I. \end{equation} $

(4.10)

Now, given any $ -\epsilon\beta \in \Lambda $, by the facts $ -\epsilon\beta \neq 0 $, $ H\subset I $ and the maximal length of $ L $ we have

$ \begin{equation} [H, L_{-\epsilon\beta\psi}] = L_{-\epsilon\beta} \subset I. \end{equation} $

(4.11)

From (4.9)-(4.11) we conclude that $ I = L $. Consequently $ L $ is simple.

Theorem 4.6 Let $ L $ be a split regular BiHom-Lie color algebra of maximal length, root multiplicative and satisfying $ \mathrm{Z}(L) = 0 $, $ H = \sum_{\alpha \in \Lambda}[L_{\alpha\psi^{-1}}, L_{-\alpha\phi^{-1}}] $. Then $ L = \oplus_{[\alpha] \in \Lambda/\sim}I_{[\alpha]}, $ where any $ I_{[\alpha]} $ is a simple $ ( $split$ ) $ ideal having its roots system $ \Lambda_{I_{[\alpha]}} $, and all of its elements are connected.

Proof  By corollary 3.7, $ L = \oplus_{[\alpha] \in \Lambda/\sim}I_{[\alpha]} $ is the direct sum of the ideals $ I_{[\alpha]} = H_{\Lambda_{\alpha}}\oplus V_{\Lambda_{\alpha}} $ = $ (\sum_{\beta \in [\alpha]}[L_{\beta\psi^{-1}}, L_{-\beta\phi^{-1}}])\oplus(\oplus_{\beta \in [\alpha]}L_{\beta}) $ having any $ I_{[\alpha]} $ its root system, $ \Lambda_{I_{[\alpha]}}: = [\alpha] $. It is easy to check that $ \Lambda_{I_{[\alpha]}} $ has all of its roots $ \Lambda_{I_{[\alpha]}} $-connected, (connected through roots in $ \Lambda_{I_{[\alpha]}} $). We also have that any of the $ I_{[\alpha]} $ is root-multiplicative as consequence of the root-multiplicativity of $ L $. Clearly $ I_{[\alpha]} $ is of maximal length, and finally $ \mathrm{Z}_{I_{[\alpha]}}(I_{[\alpha]}) $ = 0, (where $ \mathrm{Z}_{I_{[\alpha]}}(I_{[\alpha]}) $ denotes the center of $ I_{[\alpha]} $ in $ I_{[\alpha]} $), as consequence of $ [I_{[\alpha]}, I_{[\beta]}] = 0 $ if $ [\alpha]\neq [\beta] $, (Theorem 3.6), and $ \mathrm{Z}(L) = 0 $. We can apply Theorem 4.5 to any $ I_{[\alpha]} $ so as to conclude that $ I_{[\alpha]} $ is simple. It is clear that the decomposition $ L = \oplus_{[\alpha] \in \Lambda/\sim}I_{[\alpha]} $ satisfies the assertions of the theorem.