On split twisted inner derivation triple systems with no restrictions on their 0-root spaces

Definition 2.1

Let T be a triple system such that its triple product satisfies:

1. { x , y , z } = ε { y , x , z } , with ε a fixed element in ± 1 ,

2. { x , y , z } + { y , z , x } + { z , x , y } = 0 ,

3. { x , y , { a , b , c } } − { a , b , { x , y , c } } = { { x , y , a } , b , c } + { a , { x , y , b } , c } ,

Definition 2.2

A triple system T is called Jordan triple system if its triple product satisfies:

1. { x , y , z } = { z , y , x } ,

2. { x , y , { a , b , c } } − { a , b , { x , y , c } } = { { x , y , a } , b , c } − { a , { y , x , b } , c } ,

Definition 2.3

[8] Let ( T , { ⋅ , ⋅ , ⋅ } ) be a triple system. We say that T is a twisted inner derivation triple system, if there exists a linear bijection, τ , of order one or two on L ≔ span K { L ( x , y ) : x , y ∈ T } , where L ( x , y ) denotes the left multiplication operator in T , L ( x , y ) ( z ) ≔ { x , y , z } , such that

and

for any x , y , z , a , b , c ∈ T , where { a , b , c } ′ ≔ τ [ L ( a , b ) ] ( c ) .

Definition 2.4

Let I be a linear subspace of a twisted inner derivation triple system T . If { I , I , I } + { I , I , I } ′ ⊆ I , we say that I is a subsystem of T . If { I , T , T } + { T , I , T } + { T , T , I } + { I , T , T } ′ + { T , I , T } ′ ⊆ I , we say that I is an ideal of T .

Definition 2.5

The annihilator of a twisted inner derivation triple system T is the set Ann ( T ) = { x ∈ T : { x , T , T } + { T , x , T } + { T , T , x } = 0 } .

Definition 2.6

[8] The standard embedding of a twisted inner derivation triple system ( T , { ⋅ , ⋅ , ⋅ } ) is the two-graded algebra A = L ⊕ T , whose product is given by

for any x , y , z ∈ T , and l 1 , l 2 ∈ L .

Definition 2.7

[20] Let T be a twisted inner derivation triple system, A = L ⊕ T be its standard embedding, and H be an M A S A of Lie algebra L satisfying τ ( H ) ⊂ H . For a linear functional θ ∈ ( H ) ∗ , we define the root space of T (with respect to H ) associated with θ as the subspace T θ ≔ { t θ ∈ T : h t θ = θ ( h ) t θ for any h ∈ H } . The elements θ ∈ ( H ) ∗ satisfying T θ ≠ 0 are called roots of T with respect to H and we denote Λ T ≔ { θ ∈ ( H ) ∗ ⧹ { 0 } : T θ ≠ 0 } .

Definition 2.8

[20] Let T be a twisted inner derivation triple system, A = L ⊕ T be its standard embedding, and let H be an M A S A of Lie algebra L satisfying τ ( H ) ⊂ H . We shall call that T is a split twisted inner derivation triple system with respect to H if

We say that Λ T is the root system of T .

Lemma 2.9

Let T be a split twisted inner derivation triple system, for any θ ∈ Λ L , then θ τ ∈ Λ L and L θ τ = τ ( L θ ) .

Proof

Given any 0 ≠ e θ ∈ L θ and h ∈ H , one has [ h , τ ( e θ ) ] = τ ( [ τ ( h ) , e θ ] ) = τ ( θ ( τ ( h ) ) e θ ) = θ ( τ ( h ) ) τ ( e θ ) = ( θ τ ) ( h ) τ ( e θ ) , with τ ( e θ ) ≠ 0 ; therefore, θ τ ∈ Λ L and τ ( L θ ) ⊂ L θ τ . Thus, for any θ ∈ Λ L , one has ( L θ ) ⊂ τ ( L θ τ ) , as θ τ ∈ Λ L , we have L θ τ = τ ( L θ ) .□

Lemma 2.10

Let T be a split twisted inner derivation triple system with a standard embedding A = L ⊕ T . H is an M A S A of Lie algebra L satisfying τ ( H ) ⊂ H . For any θ , η , ζ ∈ Λ T ∪ { 0 } and any δ , ε ∈ Λ L ∪ { 0 } , then the following statements hold:

1. If T θ T η ≠ 0 , then T θ T η ⊂ L θ + η τ , that is, θ + η τ ∈ Λ L ∪ { 0 } .

2. If L δ T θ ≠ 0 , then L δ T θ ⊂ T δ + θ , that is, δ + θ ∈ Λ T ∪ { 0 } .

3. If T θ L δ ≠ 0 , then T θ L δ ⊂ T θ + δ τ , that is, θ + δ τ ∈ Λ T ∪ { 0 } .

4. If [ L δ , L ε ] ≠ 0 , then [ L δ , L ε ] ⊂ L δ + ε , that is, δ + ε ∈ Λ L ∪ { 0 } .

5. If { T θ , T η , T ζ } ≠ 0 , then { T θ , T η , T ζ } ⊆ T θ + η τ + ζ , that is, θ + η τ + ζ ∈ Λ T ∪ { 0 } .

6. If { T θ , T η , T ζ } ′ ≠ 0 , then { T θ , T η , T ζ } ′ ⊆ T θ τ + η + ζ , that is, θ τ + η + ζ ∈ Λ T ∪ { 0 } .

Proof

1. For any t θ ∈ T θ , t η ∈ T η , and h ∈ H , by equation (2.4), one has

   [ h , t θ t η ] = ( h t θ ) t η + t θ ( τ ( h ) t η ) = ( θ + η τ ) ( h ) t θ t η .

   Therefore, T θ T η ⊂ L θ + η τ .

2. For any e δ ∈ L δ , t θ ∈ T θ , and h ∈ H , we can write

   h = ∑ i = 1 m x i y i

   and

   e δ = ∑ j = 1 n z j u j

   with x i , y i , z j , u j ∈ T . By equations (2.1) and (2.4), one obtains

   h ( e δ t θ ) = ∑ i = 1 m ∑ j = 1 n { x i , y i , { z j , u j , t θ } } = ∑ i = 1 m ∑ j = 1 n ( { { x i , y i , z j } , u j , t θ } + { z j , { x i , y i , u j } ′ , t θ } + { z j , u j , { x i , y i , t θ } } ) = ∑ j = 1 n { h z j , u j , t θ } + ∑ j = 1 n { z j , τ ( h ) u j , t θ } + ∑ j = 1 n { z j , u j , θ ( h ) t θ } = ∑ j = 1 n ( h z j ) u j + ∑ j = 1 n z j ( τ ( h ) u j ) t θ + θ ( h ) e δ t θ = ∑ j = 1 n [ h , z j u j ] t θ + θ ( h ) e δ t θ = [ h , e δ ] t θ + θ ( h ) e δ t θ = ( δ + θ ) ( h ) e δ t θ .

   Therefore, L δ T θ ⊂ T δ + θ .

3. Consequence of Lemma 2.9 and items (2).

4. For any e δ ∈ L , e ε ∈ L , and h ∈ H , by Jacobi identity of Lie algebra L , one obtains [ h , [ e δ , e ε ] ] = [ e δ , [ h , e ε ] ] + [ [ h , e δ ] , e ε ] = [ e δ , ε ( h ) e ε ] + [ δ ( h ) e δ , e ε ] = ( δ + ε ) ( h ) [ e δ , e ε ] . Therefore, [ L δ , L ε ] ⊂ L δ + ε .

5. Consequence of items (1) and (2).

6. Consequence of items (1) and (2), and Lemma 2.9.□

Definition 2.11

Let Λ T be a root system of a split twisted inner derivation triple system T , if it satisfies that θ ∈ Λ T implies − θ , θ τ ∈ Λ T , we say that Λ T is symmetric.

A similar concept applies to the set Λ L of nonzero roots of L .

Definition 2.12

Let θ and η be two nonzero roots, we shall say that θ and η are connected if there exists a family { θ 1 , θ 2 , … , θ 2 n , θ 2 n + 1 } ⊂ Λ T ∪ { 0 } of roots of T such that

1. { θ 1 , θ 1 + θ 2 τ + θ 3 , θ 1 + θ 2 τ + θ 3 + θ 4 τ + θ 5 , … , θ 1 + θ 2 τ + ⋯ + θ 2 n τ + θ 2 n + 1 } ⊂ Λ T ;

2. { θ 1 + θ 2 τ , θ 1 + θ 2 τ + θ 3 + θ 4 τ , … , θ 1 + θ 2 τ + ⋯ + θ 2 n τ } ⊂ Λ L ;

3. θ 1 = θ and θ 1 + θ 2 τ + ⋯ + θ 2 n τ + θ 2 n + 1 ∈ { ± η , ± η τ } .

Definition 2.13

Let Ω T be a subset of a root system Λ T , if it is symmetric, and given θ , η , ζ ∈ Ω T ∪ { 0 } such that θ + η τ ∈ Λ L and θ + η τ + ζ ∈ Λ T , then θ + η τ + ζ ∈ Ω T , and we say that Ω T is a root subsystem.

Definition 2.14

Let Ω T be a root subsystem of Λ T . We define

and V Ω T ≔ ⊕ θ ∈ Ω T T θ . Taking into account the fact that { T 0 , T 0 , T 0 } = 0 , it is straightforward to verify that

is a subsystem of T . We will say that T Ω T is a twisted inner derivation triple subsystem associated with the root subsystem Ω T .

Proposition 2.15

If Λ L is symmetric, then the relation ∼ in Λ T , defined by θ ∼ η if and only if η ∈ Λ θ T , is of equivalence.

Proposition 2.16

Let θ be a nonzero root and suppose Λ L is symmetric. Then Λ θ T is a root subsystem.

Lemma 3.1

The following assertions hold:

1. Suppose θ , η ∈ Λ T , T θ T η ≠ 0 , then θ ∼ η .

2. Suppose θ , η ∈ Λ T , θ ∈ Λ L , L θ T η ≠ 0 , then θ ∼ η .

3. Suppose θ , η ∈ Λ T , θ ∈ Λ L , T η L θ ≠ 0 , then θ ∼ η .

4. Suppose θ , η ∈ Λ T , θ , η ∈ Λ L , [ L θ , L η ] ≠ 0 , then θ ∼ η .

5. Suppose θ , η ¯ ∈ Λ T such that θ is not connected with η ¯ , then T θ T η ¯ = 0 , L θ T η ¯ = 0 and T η ¯ L θ = 0 if furthermore θ ∈ Λ L . Suppose θ , η ¯ ∈ Λ T such that θ is not connected with η ¯ , [ L θ , L η ¯ ] = 0 if furthermore θ , η ¯ ∈ Λ L .

Proof

1. Suppose T θ T η ≠ 0 , by Lemma 2.10 (1), we obtain θ + η τ ∈ Λ L ∪ { 0 } . If θ + η τ = 0 , then η = − θ τ and so θ ∼ η . Suppose θ + η τ ≠ 0 , by θ + η τ ∈ Λ L , we find { θ , η , − θ } is a connection from θ to η , so θ ∼ η .

2. Suppose L θ T η ≠ 0 , by Lemma 2.10 (2), we obtain θ + η ∈ Λ T ∪ { 0 } . If θ + η = 0 , then θ ∼ η . Suppose θ + η ≠ 0 , by θ + η ∈ Λ T , we find { θ , 0 , − θ − η } is a connection from θ to η , so θ ∼ η .

3. Suppose T η L θ ≠ 0 , by Lemma 2.10 (3), we obtain η + θ τ ∈ Λ T ∪ { 0 } . If η + θ τ = 0 , then θ ∼ η . Suppose η + θ τ ≠ 0 , by η + θ τ ∈ Λ T , we find { θ , 0 , − η τ − θ } is a connection from θ to η , so θ ∼ η .

4. Suppose [ L θ , L η ] ≠ 0 , by Lemma 2.10 (4), we obtain θ + η ∈ Λ L ∪ { 0 } . If θ + η = 0 , then θ ∼ η . Suppose θ + η ≠ 0 , by θ + η ∈ Λ L , we find { θ , η τ , − θ } is a connection from θ to η , so θ ∼ η .

5. It is a consequence of items (1), (2), (3), and (4).□

Lemma 3.2

If θ , η ¯ ∈ Λ T are not connected, then

Proof

If T θ T − θ τ = 0 it is clear. One suppose that T θ T − θ τ ≠ 0 and η ¯ ( T θ T − θ τ ) ≠ 0 , one obtains [ T θ T − θ τ , L η ¯ ] = η ¯ ( T θ T − θ τ ) L η ¯ ≠ 0 . But see equation (2.4), one obtains

By Lemmas 2.9 and 3.1 (5), one obtains L η ¯ T θ = τ ( L η ¯ ) T − θ τ = 0 , that is, [ T θ T − θ τ , L η ¯ ] = 0 , a contradiction. Finally, one suppose T θ T − θ τ ≠ 0 and η ¯ ( τ ( T θ T − θ τ ) ) ≠ 0 , one obtains [ τ ( T θ T − θ τ ) , L η ¯ ] = η ¯ ( τ ( T θ T − θ τ ) ) L η ¯ ≠ 0 . But see equation (2.7), one obtains

By Lemmas 2.9 and 3.1 (5), one obtains ( τ L η ¯ ) T θ = L η ¯ T − θ τ = 0 , that is, [ τ ( T θ T − θ τ ) , L η ¯ ] = 0 , a contradiction.□

Lemma 3.3

Fix θ 0 ∈ Λ T and suppose Λ L is symmetric. For θ ∈ Λ θ 0 T and η , ζ ∈ Λ T ∪ { 0 } , then the following assertions hold:

1. If { T θ , T η , T ζ } ≠ 0 (resp., { T θ , T η , T ζ } ′ ≠ 0 ), then η , ζ , θ + η τ + ζ ∈ Λ θ 0 T ∪ { 0 } (resp., η , ζ , θ τ + η + ζ ∈ Λ θ 0 T ∪ { 0 } ).

2. If { T η , T θ , T ζ } ≠ 0 (resp., { T η , T θ , T ζ } ′ ≠ 0 ), then η , ζ , η + θ τ + ζ ∈ Λ θ 0 T ∪ { 0 } (resp., η , ζ , η τ + θ + ζ ∈ Λ θ 0 T ∪ { 0 } ).

3. If { T η , T ζ , T θ } ≠ 0 (resp., { T η , T ζ , T θ } ′ ≠ 0 ), then η , ζ , η + ζ τ + θ ∈ Λ θ 0 T ∪ { 0 } (resp. η , ζ , η τ + ζ + θ ∈ Λ θ 0 T ∪ { 0 } ).