On split involutive regular BiHom-Lie superalgebras

Definition 2.1

A BiHom-Lie superalgebra L is a Z 2 -graded algebra L = L 0 ¯ ⊕ L 1 ¯ , endowed with an even bilinear mapping [ ⋅ , ⋅ ] : L × L → L and two homomorphisms ϕ , ψ : L → L satisfying the following conditions, for all x ∈ L i ¯ , y ∈ L j ¯ , z ∈ L k ¯ and i ¯ , j ¯ , k ¯ ∈ Z 2 :

Lemma 2.2

Let ( L , ⁎ ) be a split involutive regular BiHom-Lie superalgebra. Then, for any α , β ∈ Λ ∪ {0} ,

1. ϕ ( L α ) = L α ϕ − 1 and ϕ − 1 ( L α ) = L α ϕ ;

2. ψ ( L α ) = L α ψ − 1 and ψ − 1 ( L α ) = L α ψ ;

3. [ L α , L β ] ⊂ L α ϕ − 1 + β ψ − 1 ;

4. ( L α ) ⁎ = L − α .

Lemma 2.3

The following assertions hold:

1. If α ∈ Λ, then α ϕ − z 1 ψ − z 2 ∈ Λ for any z 1 , z 2 ∈ Z ;

2. L 0 = H .

Definition 2.4

A root system Λ of a split involutive regular BiHom-Lie superalgebra is called symmetric if it satisfies that α ∈ Λ implies −α ∈ Λ.

Definition 3.1

Let α and β be two nonzero roots. We will say that α is connected to β if either

or there exists {α ₁,…,α _k } ⊂ Λ with k ≥ 2 , such that

1. α 1 ∈ { α ϕ − n ψ − r :    n , r ∈ ℕ } ;

2. α 1 ϕ − 1 + α 2 ψ − 1 ∈ Λ , α 1 ϕ − 2 + α 2 ϕ − 1 ψ − 1 + α 3 ψ − 1 ∈ Λ , α 1 ϕ − 3 + α 2 ϕ − 2 ψ − 1 + α 3 ϕ − 1 ψ − 1 + α 4 ψ − 1 ∈ Λ , ⋯ α 1 ϕ − i + α 2 ϕ − i + 1 ψ − 1 + α 3 ϕ − i + 2 ψ − 1 + ⋯ + α i ϕ − 1 ψ − 1 + α i + 1 ψ − 1 ∈ Λ , α 1 ϕ − k + 2 + α 2 ϕ − k + 3 ψ − 1 + α 3 ϕ − k + 4 ψ − 1 + ⋯ + α k − 2 ϕ − 1 ψ − 1 + α k − 1 ψ − 1 ∈ Λ ;

3. α 1 ϕ − k + 1 + α 2 ϕ − k + 2 ψ − 1 + α 3 ϕ − k + 3 ψ − 1 + ⋯ + α i ϕ − k + i ψ − 1 + ⋯ + α k − 1 ϕ − 1 ψ − 1 + α k ψ − 1 ∈ { ± β ϕ − m ψ − s : m , s ∈ ℕ } .

We will also say that {α ₁,…,α _k } is a connection from α to β.

Proposition 3.2

The relation ∼ in Λ, defined by α ∼ β if and only if α is connected to β, is an equivalence relation.

Proposition 3.3

For any [α] ∈ Λ/∼, the following assertions hold:

1. [I _[α], I _[α]] ⊂ I _[α];

2. ϕ(I _[α]) = I _[α] and ψ(I _[α]) = I _[α];

3. I [ α ] ⁎ = I [ α ] ;

4. for any [β] ≠ [α], we have [I _[α], I _[β]] = 0.

Proof

1. First, we check that [I _[α], I _[α]] ⊂ I _[α], and we can write

   (3.1) [ I [ α ] , I [ α ] ] = [ I H , [ α ] ⊕ V [ α ] , I H , [ α ] ⊕ V [ α ] ] ⊂ [ I H , [ α ] , V [ α ] ] + [ V [ α ] , I H , [ α ] ] + [ V [ α ] , V [ α ] ] .

   Given β ∈ [α], we have [ I H , [ α ] ] ⊂ L β ψ − 1 . Since βψ ⁻¹ ∈ [α], it follows that [ I H , [ α ] , L β ] ⊂ V [ α ] .

   By a similar argument, we get [ L β , I H , [ α ] ] ⊂ V [ α ] .

   Next we consider [V _[α], V _[α]]. If we take β, γ ∈ [α] and i ¯ , j ¯ ∈ Z 2 such that [ L β , i ¯ , L γ , j ¯ ] ≠ 0 , then [ L β , L γ ] ⊂ L β ϕ − 1 + γ ψ − 1 , i ¯ + j ¯ . If βϕ ⁻¹ + γψ ⁻¹ = 0, we get [ L β , L γ ] = [ L β , L − β ] = [ L β , ( L β ) ⁎ ] ⊂ H and so [ L β , L γ ] ⊂ I H , [ α ] . Suppose that βϕ ⁻¹ + γψ ⁻¹ ∈ Λ. We infer that {β, γ} is a connection from β to βϕ ⁻¹ + γψ ⁻¹. The transitivity of ∼ shows that βϕ ⁻¹ + γψ ⁻¹ ∈ [α] and so [ L β , L γ ] ⊂ V [ α ] . Hence,

   (3.2) [ V [ α ] , V [ α ] ] ∈ I [ α ] .

   From (3.1) and (3.2), we get [I _[α], I _[α]] ⊂ I _[α].

2. It is easy to check that ϕ(I _[α]) = I _[α] and ψ(I _[α]) = I _[α].

3. It is easy to check that I [ α ] ⁎ = I [ α ] .

4. For the expression [I _[α], I _[β]], we first note that

First, we consider the summand [V _[α],V _[β]] and suppose that there exist α ₁ ∈ [α], β ₁ ∈ [β] and j ¯ , k ¯ ∈ Z 2 such that [ L α 1 , j ¯ , L β 1 , k ¯ ] ≠ 0 . By the hypothesis, α ₁ ϕ ⁻¹ ≠ −β ₁ ψ ⁻¹, and thus α ₁ ϕ ⁻¹ + β ₁ ψ ⁻¹ ∈ Λ. So {α ₁, β ₁, −α ₁ ϕ ⁻² ψ} is a connection between α ₁ and β ₁. By the transitivity of the connection relation we see α ∈ [β], a contradiction. Hence, [ L α 1 , j ¯ , L β 1 , k ¯ ] = 0 , and so

Next we consider the first summand [I _H,[α],V _[β]] on the right-hand side of (3.3). Suppose that there exist β ∈ [α] and η ∈ [β] such that

If

then BiHom-Jacobi superidentity gives

Hence, there exists i ¯ , j ¯ , k ¯ ∈ Z 2 such that [ ψ 2 ( L η , i ¯ ) , [ ψ ( L − β , j ¯ ) , ϕ ( L β , k ¯ ) ] ] ≠ 0 . We get either [ ψ ( L β , k ¯ ) , ϕ ( L η , i ¯ ) ] ≠ 0 or [ ψ ( L η , i ¯ ) , ϕ ( L − β , j ¯ ) ] ≠ 0 , which contradicts (3.4). Therefore, [ [ L β ψ − 1 , L − β ϕ − 1 ] , ϕ 2 ( L η ) ] = 0 . Consequently, [I _H,[α],V _[β]] = 0. In a similar way, one can prove [V _[α],I _H,[β]] = 0 and the proof is completed.□

Theorem 3.4

The following assertions hold:

1. For any [α] ∈ Λ/∼, the involutive subalgebra I _[α] = I _H,[α] + V _[α] of L associated with [α] is an involutive ideal of L .

2. If L is simple, then there exists a connection from α to β for any α, β ∈ Λ and H = ∑ α ∈ Λ ( [ L α ψ − 1 , ( L α ϕ − 1 ) ⁎ ] ) .

Proof

(1) Since [I _[α], H] ⊂ I _[α], by Lemmas 2.2 and 2.3, we have

According to Propositions 3.2 and 3.3, we have

As we also have ϕ(I _[α]) = I _[α] and ψ(I _[α]) = I _[α], we conclude that I _[α] is an ideal of L .

(2) The simplicity of L implies I [ α ] = L , then [α] = Λ and H = ∑ α ∈ Λ ( [ L α ψ − 1 , ( L α ϕ − 1 ) ⁎ ] ) .□

Theorem 3.5

We have

where U is a linear complement in H of span K { [ L α ψ − 1 , ( L α ϕ − 1 ) ⁎ ] : α ∈ Λ } and any I _[α] is one of the involutive ideals of L described in Theorem 3.4, satisfying [I _[α],I _[β]] = 0 if [α] ≠ [β].

Proof

I _[α] is well defined and is an ideal of L , being clear that

Finally, Proposition 3.3 gives us [I _[α], I _[β]] = 0 if [α] ≠ [β].□

Corollary 3.6

If Z ( L ) = 0 and H = ∑ α ∈ Λ ( [ L α ψ − 1 , ( L α ϕ − 1 ) ⁎ ] ) , then L is the direct sum of the involutive ideal given in Theorem 3.4:

Furthermore, [I _[α], I _[β]] = 0 if [α] ≠ [β].

Proof

Since H = ∑ α ∈ Λ ( [ L α ψ − 1 , ( L α ϕ − 1 ) ⁎ ] ) , it follows that L = ∑ [ α ] ∈ Λ / ∼ I [ α ] . To verify the direct character of the sum, take some v ∈ I [ α ] ∩ ( ∑ [ β ] ∈ Λ / ∼ , [ β ] ≠ [ α ] I [ β ] ) . Since v ∈ I _[α], the fact [I _[α], I _[β]] = 0 when [α] ≠ [β] gives us

In a similar way, v ∈ ∑ [ β ] ∈ Λ / ∼ , [ β ] ≠ [ α ] I [ β ] implies [v, I _[α]] = 0. That is, v ∈ Z ( L ) and so v = 0.□

Lemma 4.1

Let L be a split regular BiHom-Lie superalgebra with Z ( L ) = 0 and I an ideal of L . If I ⊂ H, then I = {0}.

Proof

It is analogous to [12].□

Definition 4.2

We say that a split involutive regular BiHom-Lie superalgebra L is root-multiplicative if given α ∈ Λ i ¯ , β ∈ Λ j ¯ , for i ¯ , j ¯ ∈ Z 2 , such that α ψ − 1 + β ϕ − 1 ∈ Λ i ¯ + j ¯ , then [ L α , i ¯ , L β , j ¯ ] ≠ 0 .

Definition 4.3

A split involutive regular BiHom-Lie superalgebra L is of maximal length if dim L α , i ¯ = 1 for any α ∈ Λ and i ¯ ∈ Z 2 .

Observe that if L is of maximal length, then we have

where Λ i ¯ I = { α ∈ Λ : I i ¯ ∩ L α , i ¯ ≠ 0 } , i ¯ ∈ Z 2 .

Definition 4.4

A BiHom-Lie superalgebra L is called perfect if Z ( L ) = 0 and [ L , L ] = L .

Proposition 4.5

Let L be a perfect split involutive regular BiHom-Lie superalgebra of maximal length and root-multiplicative. If L has all of its roots connected, then any ideal I of L satisfies I* = I.

Proof

Let I be a nonzero ideal of L , and we can write

with Λ i ¯ I ⊂ Λ i ¯ , i ¯ ∈ Z 2 , and some Λ i ¯ I ≠ ∅ . Hence, let us fix some α 0 ∈ Λ i ¯ I such that

By Lemma 2.2, ϕ ( L α 0 ) = L α 0 ϕ − 1 and ψ ( L α 0 ) = L α 0 ψ − 1 . By (4.3), we have ϕ ( L α 0 , i 0 ¯ ) ⊂ ϕ ( I ) = I and ψ ( L α 0 , i 0 ¯ ) ⊂ ψ ( I ) = I . So L α 0 ϕ − 1 ψ − 1 , i 0 ¯ ⊂ I . Similarly, we get

For any β ∈ Λ, β ∉ ±α ₀ ϕ ⁻ⁿ⁺¹ ψ ⁻¹, for n ∈ ℕ , the fact that α ₀ and β are connected gives us a connection {γ ₁,γ ₂,…,γ _n } ⊂ Λ from α ₀ to β such that

Taking into account that γ ₁, γ ₂ and γ ₁ ϕ ⁻¹ + γ ₂ ψ ⁻¹. Since γ 1 = α 0 ∈ Λ i 0 ¯ and γ 2 ∈ Λ i 2 ¯ , the root-multiplicativity and maximal length of L show γ 1 ϕ − 1 + γ 2 ψ − 1 ∈ Λ i ¯ + j ¯ .

and by (4.4), we have

We can argue in a similar way from γ ₁ ϕ ⁻¹ + γ ₂ ψ ⁻¹, γ ₃ and γ ₁ ϕ ⁻² + γ ₂ ϕ ⁻¹ ψ ⁻¹, γ ₃ ψ ⁻¹. Hence,

and by above, we have

Following this process with the connection {γ ₁,γ ₂,…,γ _n }, we obtain that

And so that we get either

for any β ∈ Λ , m ∈ ℕ . From Lemma 2.2, we get

In both cases

And so [e _β ,e _−β ] ∈ I. As given any e − α 0 ∈ L − α 0 , we have

we conclude L − α 0 ⊂ I , and then we get ( L α 0 ) ⁎ ⊂ I . Hence, ( ⊕ α ∈ Λ I L α ) ⁎ = ⊕ α ∈ Λ I L α .

Finally, the fact H = ∑ α ∈ Λ ( [ L α ψ − 1 , L − α ϕ − 1 ] ) and (4.7) give us

As H* = H, we get (I ∩ H)* = (I ∩ H), taking into account ( ⊕ α ∈ Λ I L α ) ⁎ = ⊕ α ∈ Λ I L α . The decomposition of I in (4.1) finally gives us I* = I.□

Theorem 4.6

Let L be a perfect split involutive regular BiHom-Lie superalgebra of maximal length and root-multiplicative. Then L is simple if and only if it has all its nonzero roots connected.

Proof

The first implication is Theorem 3.4. To prove the converse, write L = H + ∑ [ α ] ∈ Λ / ∼ L [ α ] and consider I a nonzero ideal of L . By (4.8), we have H ⊂ I. Given any α ∈ Λ and taking into account α ≠ 0 and the maximal length of L , we have L α ⊂ I . We conclude I = L , and therefore L is simple.□

Theorem 4.7

Let L be a perfect split involutive regular BiHom-Lie superalgebra of maximal length and root-multiplicative. Then L is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular BiHom-Lie superalgebra and having all its nonzero roots connected.

Proof

By Corollary 3.6, L = ⊕ [ α ] ∈ Λ / ∼ I [ α ] is the direct sum of the family of ideals

where each I _[α] is a split involutive regular BiHom-Lie superalgebra with root system L I [ α ] = [ α ] . To make use of Theorem 4.6 in each I _[α], we observe that the root-multiplicativity of L and Proposition 3.3 show that L I [ α ] has all of its elements L I [ α ] connected, that is, connected through connections contained in L I [ α ] . Moreover, each I _[α] is root-multiplicative by the root-multiplicativity of L . So we infer that I _[α] is of maximal length and finally its center Z(I _[α]) = 0. As a consequence, [I _α , I _β ] = 0 if [α] ≠ [β]. Applying Theorem 4.6, we conclude that I _[α] is simple and L = ⊕ [ α ] ∈ Λ / ∼ I [ α ] .□