A Realization of Hom-Lie Algebras by Iso-Deformed Commutator Bracket

Abstract

We construct classical Iso-Lie and Iso-Hom-Lie algebras in 𝑔𝑙(𝑉) by twisted commutator bracket through Iso-deformation. We prove that they are simple. Their Iso-automorphisms and isotopies are also presented.

1. Introduction

The conventional Lie theory has been developed in mathematical literature in its linear, local, and canonical formulation and the simplest conceivable product [𝐴,𝐵]=𝐴𝐵−𝐵𝐴, where 𝐴𝐵 is the trivial associative product. But it is not applicable to a growing number of nonlinear, nonlocal, and noncanonical systems which have recently emerged in mathematics. Lie-isotopic theory [1–3] is the generalization of the unit 𝐼 to a new one 𝐼̂ called isotopic unit or isounit. The resulting new mathematical structures include the old ones as special cases.

Hom-Lie algebras (𝔤,[⋅,⋅],𝛼) were first introduced by Hartwig et al. in [4] to describe the structure of deformation of the Witt and Virasoro algebra in 2006. They are a generalization of Lie algebras. When 𝛼=𝑖𝑑, the Hom-Lie algebras degenerate to exactly the Lie algebras. Because Hom-Lie algebras are closely related to discrete and deformed vector fields, differential calculus [5, 6], they have been researched extensively these years [7–9].

An elementary but important property of Lie algebras is that each associative algebra gives rise to a Lie algebra via the commutator bracket. But it is not natural to define Hom-Lie algebras by this way. In [10], Makhlouf and Silvestrov introduced the notion of a Hom-associative algebra (𝔤,𝜇,𝛼), in which the binary operation 𝜇 satisfies an 𝛼 twisted version of associativity. The authors mentioned that Lie-Santilli isotopies [1–3] can be modified appropriately to suit the Hom-algebras context. The relations between Hom-Lie algebras and Santilli's deformed bracket products are certainly interesting. This method solves the question how can commutator products in a linear space be modified to yield as general as possible (Hom-)Lie algebras.

The structure of Lie algebras over ℂ is better understood now [11, 12]. Classical simple Lie algebras over ℂ can be classified into 𝐴, 𝐵, 𝐶, 𝐷 types. Similarly, we can use the twisted commutator bracket through Iso-deformation to study classical Iso-(Hom-)Lie algebras over ℂ.

The paper is organized as follows: in Section 2, we study relations between multiplicative Hom-Lie algebras and Lie algebras; in Section 3, we construct Iso-Lie and Iso-Hom-Lie algebras; in Section 4, classical II types 𝐴, 𝐵, 𝐶, 𝐷-Iso-Lie and Iso-Hom-Lie algebras are studied. We prove that they are simple. Their Iso-automorphism and isotopy are also presented.

Throughout this paper, we denote by “𝐸” the identity matrix, “𝑔𝐿(𝑉)” the linear group, and “𝑥^𝑡” the transpose of matrix 𝑥, and “≅” denotes the isomorphism of algebras. Without otherwise stated, all algebras are finite dimensional and over the complex field ℂ.

2. Hom-Lie Algebras

Definition 1 (see [4]). A Hom-Lie algebra is a triple (𝔤,[⋅,⋅],𝛼) consisting of a vector space 𝔤 over ℂ, a linear sef-map 𝛼, and a bilinear map [⋅,⋅]∶𝔤×𝔤→𝔤 satisfying 𝛼(𝑥),+𝛼𝑦,[+𝛼(𝑧),𝑥,𝑦=−𝑦,𝑥,∀𝑥,𝑦∈𝔤;𝑦,𝑧𝑧,𝑥]𝑥,𝑦=0,∀𝑥,𝑦,𝑧∈𝔤. (1)

When 𝛼=𝑖𝑑, (𝔤,[⋅,⋅],𝛼) is degenerated to Lie algebra (𝔤,[⋅,⋅]). A Hom-Lie algebra (𝔤,[⋅,⋅],𝛼) is called multiplicative if 𝛼𝑥,𝑦=𝛼(𝑥),𝛼𝑦,∀𝑥,𝑦∈𝔤 (2) is satisfied [13]. We call (𝔤,[⋅,⋅],𝛼) Lie-type if there exists a Lie algebra (𝔤,[⋅,⋅]^′) such that [𝑥,𝑦]=𝛼([𝑥,𝑦]^′)=[𝛼(𝑥),𝛼(𝑦)]^′,∀𝑥,𝑦∈𝔤. At the same time (𝔤,[⋅,⋅]^′) is called the Hom-Lie admissible algebra of (𝔤,[⋅,⋅],𝛼).

A subspace 𝔤₁ of 𝔤 is called an ideal of (𝔤,[⋅,⋅],𝛼) if 𝛼(𝔤₁)⊆𝔤₁, [𝔤₁,𝔤]⊆𝔤₁ are satisfied. We call a Hom-Lie algebra (𝔤,[⋅,⋅],𝛼) simple if it has no nontrivial ideals and [𝔤,𝔤]=𝔤.

An automorphism 𝜓 of a Hom-Lie algebra (𝔤,[⋅,⋅],𝛼) is an invertible linear self-map satisfying 𝜓([𝑥,𝑦])=[𝜓(𝑥),𝜓(𝑦)], ∀𝑥,𝑦∈𝔤 and 𝜓𝛼=𝛼𝜓.

An isomorphism of Hom-Lie algebras (𝔤₁,[⋅,⋅]₁,𝛼₁) with (𝔤₂,[⋅,⋅]₂,𝛼₂) is an invertible linear map 𝜙∶𝔤₁→𝔤₂ satisfying 𝜙([𝑥,𝑦]₁)=[𝜙(𝑥),𝜙(𝑦)]₂,∀𝑥,𝑦∈𝔤₁ and 𝛼₂𝜙=𝜙𝛼₁.

Lemma 2. Let (𝔤,[⋅,⋅],𝛼) be a multiplicative simple Hom-Lie algebra. Then 𝛼 is invertible.

Proof. It is easy to check that Ker(𝛼) is an ideal of (𝔤,[⋅,⋅],𝛼). By the simplicity of (𝔤,[⋅,⋅],𝛼), we have Ker(𝛼)=0. That is, 𝛼 is invertible.

Lemma 3 (see [14]). Let (𝔤,[⋅,⋅]) be a Lie algebra over ℂ with an algebraic homomorphism 𝛼. Define a bracket [⋅,⋅]_𝛼 by [𝑥,𝑦]_𝛼=𝛼([𝑥,𝑦]), ∀𝑥,𝑦∈𝔤. Then (𝔤,[⋅,⋅]_𝛼,𝛼) is a Hom-Lie algebra.

Theorem 4. Let (𝔤,[⋅,⋅],𝛼) be a multiplicative Hom-Lie algebra with 𝛼 invertible. Then (𝔤,[⋅,⋅],𝛼) is Lie-type with the Hom-Lie admissible algebra (𝔤,[⋅,⋅]^′), where [⋅,⋅]^′ is defined by [𝑥,𝑦]^′=𝛼⁻¹([𝑥,𝑦]),∀𝑥,𝑦∈𝔤.

Proof. Define [𝑥,𝑦]^′=𝛼⁻¹([𝑥,𝑦]),∀𝑥,𝑦∈𝔤. It is easy to check that 𝛼⁻¹𝑥,𝑦^′=𝛼⁻¹(𝑥),𝛼⁻¹𝑦^′,∀𝑥,𝑦∈𝔤. (3) Now we prove that (𝔤,[⋅,⋅]^′) is a Lie algebra. The skew-symmetricity of [⋅,⋅]^′ is obvious. ∀𝑥,𝑦,𝑧∈𝔤, ↺_{𝑥,𝑦,𝑧}𝑥,𝑦,𝑧^′′=↺_{𝑥,𝑦,𝑧}𝛼⁻¹𝑥,𝛼⁻¹𝑦,𝑧=↺_{𝑥,𝑦,𝑧}𝛼⁻¹𝛼⁻¹𝛼(𝑥),𝛼⁻¹𝑦,𝑧=↺_{𝑥,𝑦,𝑧}𝛼⁻²𝛼(𝑥),𝑦,𝑧=0, (4) here ↺_{𝑥,𝑦,𝑧} denotes the cyclic summation over 𝑥,𝑦,𝑧. We have the conclusion.

Lemma 5 (see [15]). Let (𝔤₁,[⋅,⋅]₁,𝛼) and (𝔤₁,[⋅,⋅]₂,𝛽) be Lie type Hom-Lie algebra with 𝛽 invertible. Then an invertible linear map 𝜙 is an isomorphism of Hom-Lie algebras if and only if 𝜙 is an isomorphism of their Hom-Lie admissible algebras satisfying 𝛽𝜙=𝜙𝛼.

Lemma 6 (see [15]). Let (𝔤,[⋅,⋅],𝛼) be a Lie type Hom-Lie algebra with 𝛼 invertible. Then 𝜓 is an automorphism of (𝔤,[⋅,⋅],𝛼) if and only if 𝜓 is an automorphism of the Hom-Lie admissible algebra satisfying 𝛼𝜓=𝜓𝛼.

Proposition 7. Let (𝔤,[⋅,⋅],𝛼) be a finite dimensional Lie type Hom-Lie algebra with 𝛼 invertible. If its Hom-Lie admissible algebra is simple, then (𝔤,[⋅,⋅],𝛼) is simple.

Proof. By Theorem 4, the Hom-Lie admissible algebra can be written as (𝔤,[⋅,⋅]^′), where [⋅,⋅]^′ is defined by [𝑥,𝑦]^′=𝛼⁻¹([𝑥,𝑦]),∀𝑥,𝑦∈𝔤. Suppose [𝔤,𝔤]⫋𝔤, then [𝔤,𝔤]^′=𝛼⁻¹([𝔤,𝔤])=𝛼⁻¹(𝔤),𝛼⁻¹(𝔤)⊆[𝔤,𝔤]⫋𝔤, (5) which is a contradiction with the simplicity of (𝔤,[⋅,⋅]^′). This reduces to [𝔤,𝔤]=𝔤.
Let 𝔤₁ be a nontrivial ideal of (𝔤,[⋅,⋅],𝛼). By definition there are 𝛼(𝔤₁)⊆𝔤₁; [𝔤₁,𝔤]⊆𝔤₁. Therefore, [𝔤₁,𝔤]^′=𝛼⁻¹([𝔤₁,𝔤])⊆𝛼⁻¹(𝔤₁)⊆𝔤₁. That is, 𝔤₁ is a nontrivial ideal of the Hom-Lie admissible algebra, which is impossible. We have the conclusion.

3. Iso Algebras

Let 𝔤 be an associative algebra with the conventional associative product 𝑥𝑦,∀𝑥,𝑦∈𝔤 over ℂ. An Iso-associative algebra (𝔤,∗) is the same vector space 𝔤 over ℂ with a product ∗ satisfying (𝑥∗𝑦)∗𝑧=𝑥∗(𝑦∗𝑧),∀𝑥,𝑦,𝑧∈𝔤, where the product ∗ is defined by one of the following cases, the parameters 𝑇, 𝑤 appearing in cases II–V are fixed not necessary belonging to 𝔤 (it deserves to mention that in [3], (𝔤,∗) is an Iso-associative algebra over an Iso-field ℂ, specially, in this paper we study Iso-algebra over ℂ): I:  𝑥∗𝑦=𝑎𝑥𝑦,0≠𝑎∈ℂis  fixed; II:  𝑥∗𝑦=𝑥𝑇𝑦,∀𝑥,𝑦∈𝔤,𝑇is  invertible; III:  𝑥∗𝑦=𝑤𝑥𝑤𝑦𝑤,∀𝑥,𝑦∈𝔤,𝑤²=𝑤≠0; IV:  𝑥∗𝑦=𝑤𝑥𝑤𝑇𝑤𝑦𝑤,∀𝑥,𝑦∈𝔤,𝑤²=𝑤,𝑤,𝑇≠0; V:  𝑥∗𝑦=𝑎𝑤𝑥𝑤𝑇𝑤𝑦𝑤,𝑥,𝑦∈𝔤,𝑎∈ℂ,𝑤²=𝑤;𝑎,𝑤,𝑇≠0.It is obvious that of the above Iso-associative algebras only the first four are independent and there is no essential difference between type I and the conventional associative algebra.

Definition 8. An Iso-Lie algebra (𝔤,[⋅,⋅]) is a vector space 𝔤 over ℂ with a bracket [⋅,⋅] defined by [𝑥,𝑦]=𝑥∗𝑦−𝑦∗𝑥,∀𝑥,𝑦∈𝔤, where (𝔤,∗) is an Iso-associative algebra. At the same time (𝔤,∗) is called the Lie admissible algebra of (𝔤,[⋅,⋅]).

We call an isomorphism of Iso-algebras an isotopy. An automorphism of an Iso-algebra is called an Iso-automorphism.

We call (𝔤,[⋅,⋅],𝛼) an Iso-Hom-Lie algebra if its Hom-Lie admissible algebra is an Iso-Lie algebra.

Corollary 9. Let (𝔤,[⋅,⋅]) be an Iso-Lie algebra with an Iso-automorphism 𝛼. Define a bracket [⋅,⋅]_𝛼 by [𝑥,𝑦]_𝛼=𝛼([𝑥,𝑦]),∀𝑥,𝑦∈𝔤. Then (𝔤,[⋅,⋅]_𝛼,𝛼) is an Iso-Hom-Lie algebra.

We call (𝔤,[⋅,⋅]) I–V type Iso-Lie algebras if their Lie admissible algebras are I–V type Iso-associative algebras, respectively. And call the Iso-Hom-Lie algebras constructed in Corollary 9 I–V type Iso-Hom-Lie algebras if their Hom-Lie admissible algebras are I–V type Iso-Lie algebras, respectively. In this paper we put emphasis on type II Iso-(Hom-)Lie algebras.

4. Classical Type II Iso-(Hom-)Lie Algebras

In this section we study classical types II 𝐴_𝑛,  𝐵_𝑙,  𝐶_𝑛, 𝐷_𝑙-Iso-(Hom-)Lie algebras over ℂ. Give their Iso-automorphisms and isotopies. Prove that they are simple.

Let 𝑉 be a finite vector space, 𝑔𝑙(𝑉) is the linear space with the conventional associative product 𝑥𝑦,∀𝑥,𝑦∈𝑔𝑙(𝑉). 𝑇∈𝑔𝑙(𝑉) is an invertible element, 𝔤_𝑇⊆𝑔𝑙(𝑉). (𝔤_𝑇,[⋅,⋅]_𝑇) is an II type Iso-Lie algebra over ℂ, with the bracket defined by [𝑥,𝑦]_𝑇=𝑥𝑇𝑦−𝑦𝑇𝑥,∀𝑥,𝑦∈𝔤_𝑇. It is easy to know that the conventional linear Lie algebra over ℂ is a special II type Iso-Lie algebra when 𝑇 is degenerated to the trivial identity.

Theorem 10. Let 𝑠𝑙(𝑛+1,ℂ)_𝑇 be a vector space satisfying 𝑠𝑙𝑛+1,ℂ_𝑇=𝑥∈𝑔𝑙(𝑉)∣𝑇𝑟(𝑇𝑥)=0. (6)(1) Define a bracket [⋅,⋅]_𝑇 by [𝑥,𝑦]_𝑇=𝑥𝑇𝑦−𝑦𝑇𝑥,∀𝑥,𝑦∈𝑠𝑙(𝑛+1,ℂ)_𝑇. Then (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇) is an II type Iso-Lie algebra (one calls it an 𝐴_𝑛-Iso-Lie algebra).  (2) Define an invertible linear map 𝛼_𝑃 by 𝛼_𝑃(𝑥)=𝑃⁻¹𝑥𝑃,∀𝑥∈𝑠𝑙(𝑛+1,ℂ)_𝑇, (7) where 𝑃∈𝑔𝐿(𝑉) satisfying 𝑃𝑇=𝑇𝑃, and then 𝛼_𝑃 is an Iso-automorphism of (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇).(3) There is an isotopy between the conventional 𝐴_𝑛-type Lie algebra (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]) and (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇). Moreover, (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇) is simple.(4) Let (𝑠𝑙(𝑛+1,ℂ)_𝑇1,[⋅,⋅]_𝑇1) and (𝑠𝑙(𝑛+1,ℂ)_𝑇2,[⋅,⋅]_𝑇2) be 𝐴_𝑛-Iso-Lie algebras, and then they are isotopic.(5) Define a new bracket [⋅,⋅]_𝑇𝛼𝑃 by 𝑥,𝑦_𝑇𝛼𝑃=𝛼_𝑃𝑥,𝑦_𝑇=𝑃⁻¹𝑥∀𝑥𝑇𝑦−𝑦𝑇𝑃,𝑥,𝑦∈𝑠𝑙𝑛+1,ℂ_𝑇. (8) Then (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃) is a simple Iso-Hom-Lie algebra (one calls it an 𝐴_𝑛-Iso-Hom-Lie algebra) with an Iso-automorphism 𝜓 defined by 𝜓(𝑥)=𝑄⁻¹𝑥𝑄,∀𝑥∈𝑠𝑙(𝑛+1,ℂ)_𝑇, where 𝑄∈𝑔𝐿(𝑉) satisfying 𝑇𝑄=𝑄𝑇 and 𝑃𝑄=𝑘𝑄𝑃, for some 𝑘∈ℂ. (6) Let (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) be a Hom-Lie algebra with 𝛼_𝑄(𝑥)=𝑄⁻¹𝑥𝑄;𝑥,𝑦_𝛼𝑄=𝑄⁻¹().𝑥𝑦−𝑦𝑥𝑄,∀𝑥,𝑦∈𝑠𝑙𝑛+1,ℂ (9) If 𝑄=𝑘𝑃, for some 𝑘∈ℂ is satisfied, then (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) is isotopic to (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃). (7) Let (𝑠𝑙(𝑛+1,ℂ)_𝑇1,[⋅,⋅]_{𝑇1𝛼𝑃},𝛼_𝑃) and (𝑠𝑙(𝑛+1,ℂ)_𝑇2,[⋅,⋅]_{𝑇2𝛼𝑄},𝛼_𝑄) be 𝐴_𝑛-Iso-Hom-Lie algebras. They are isotopic if and only if 𝑃 is conjugate with 𝑘𝑄 (for some 𝑘∈ℂ).

Proof. (1) ∀𝑥,𝑦∈𝑠𝑙(𝑛+1,ℂ)_𝑇, there are 𝑇𝑟(𝑇𝑥)=0; 𝑇𝑟(𝑇𝑦)=0, then 𝑇𝑟(𝑇[𝑥,𝑦]_𝑇)=𝑇𝑟(𝑇𝑥𝑇𝑦−𝑇𝑦𝑇𝑥)=0; therefore, [𝑥,𝑦]_𝑇∈𝑠𝑙(𝑛+1,ℂ)_𝑇. The Jacobi identity can be checked directly. By Definition 8, (𝑠𝑙(𝑛+1,ℂ)_𝑇,  [⋅,⋅]_𝑇) is an II-type Iso-Lie algebra.
(2) Because 𝑇𝑃=𝑃𝑇, so 𝑇𝑃⁻¹=𝑃⁻¹𝑇 and 𝑃⁻¹𝑇𝑃=𝑇 are satisfied. Consider ∀𝑥∈𝑠𝑙(𝑛+1,ℂ)_𝑇, 𝑇𝑟𝑇𝛼_𝑃(𝑥)=𝑇𝑟𝑇𝑃⁻¹𝑥𝑃=𝑇𝑟𝑃⁻¹𝑇(𝑇𝑥)𝑥𝑃=𝑇𝑟=0. (10) Therefore 𝛼_𝑃(𝑠𝑙(𝑛+1,ℂ)_𝑇)⊆𝑠𝑙(𝑛+1,ℂ)_𝑇.  ∀𝑥,𝑦∈𝑠𝑙(𝑛+1,ℂ)_𝑇, 𝛼_𝑃𝑥,𝑦_𝑇=𝑃⁻¹𝑥𝑇𝑦−𝑦𝑇𝑥𝑃=𝑃⁻¹𝑥𝑃𝑃⁻¹𝑇𝑃𝑃⁻¹𝑦𝑃−𝑃⁻¹𝑦𝑃𝑃⁻¹𝑇𝑃𝑃⁻¹𝑥𝑃=𝑃⁻¹𝑥𝑃𝑇𝑃⁻¹𝑦𝑃−𝑃⁻¹𝑦𝑃𝑇𝑃⁻¹=𝛼𝑥𝑃_𝑃(𝑥),𝛼_𝑃𝑦_𝑇. (11) So 𝛼_𝑃 is an Iso-automorphism of (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇).
(3) Define an invertible linear map 𝜙 by 𝜙(𝑥)=𝑇⁻¹𝑥,∀𝑥∈𝑠𝑙(𝑛+1,ℂ). (12) It is obvious that 𝜙 is invertible and 𝑇𝑟(𝑇𝜙(𝑥))=0; therefore, 𝜙(𝑠𝑙(𝑛+1,ℂ))⊆𝑠𝑙(𝑛+1,ℂ)_𝑇. 𝜙(𝑥),𝜙𝑦_𝑇=𝜙(𝑥)𝑇𝜙𝑦−𝜙𝑦𝑇𝜙(𝑥)=𝑇⁻¹𝑥𝑇𝑇⁻¹𝑦−𝑇⁻¹𝑦𝑇𝑇⁻¹𝑥=𝑇⁻¹,().𝑥𝑦−𝑦𝑥=𝜙𝑥,𝑦∀𝑥,𝑦∈𝑠𝑙𝑛+1,ℂ (13) We have the first conclusion.
Suppose 𝔤_𝑇 is a nontrivial ideal of (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇). Let 𝔤=𝑇𝔤_𝑇=𝜙⁻¹(𝔤_𝑇). Then [𝔤,𝑠𝑙(𝑛+1,ℂ)]=𝜙⁻¹𝜙(𝔤),𝜙(𝑠𝑙(𝑛+1,ℂ))⊆𝜙⁻¹𝔤_𝑇,𝑠𝑙𝑛+1,ℂ_𝑇𝑇⊆𝜙⁻¹𝔤_𝑇=𝔤. (14) Therefore, 𝔤 is a nontrivial ideal of (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]), which is a contradiction with the simplicity of (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]). We have that (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇) is simple.
(4) Define an invertible linear map 𝜙 by 𝜙(𝑥)=𝑥𝑇₁𝑇₂⁻¹,∀𝑥∈𝑠𝑙𝑛+1,ℂ_𝑇1; (15) then 𝜙(𝑥)∈𝑠𝑙𝑛+1,ℂ_𝑇2,𝜙𝑥,𝑦_𝑇1=𝜙(𝑥),𝜙𝑦_𝑇2,∀𝑥,𝑦∈𝑠𝑙𝑛+1,ℂ_𝑇1 (16) can be checked as in (3) of the theorem. We have the conclusion.
(5) According to (2) and (3) of the theorem, Lemma 3, Proposition 7, and the definition of Iso-Hom-Lie algebra, we have that (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃) is an simple Iso-Hom-Lie algebra. If 𝜓 is an Iso-automorphism of (𝑠𝑙(𝑛+1,ℂ)_𝑇,  [⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃), by Lemma 6, 𝜓 is an Iso-automorphism of the Hom-Lie admissible algebra (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇) and satisfying 𝜓𝛼_𝑃=𝛼_𝑃𝜓. (17)
By (2) of the theorem again we have 𝜓(𝑥)=𝑄⁻¹𝑥𝑄,∀𝑥∈𝑠𝑙(𝑛+1,ℂ)_𝑇, where 𝑄∈𝑔𝐿(𝑉) satisfies 𝑄𝑇=𝑇𝑄. Consider ∀𝑥∈𝑠𝑙(𝑛+1,ℂ)_𝑇, and (17) is equivalent to 𝜓𝛼_𝑃(𝑥)=𝛼_𝑃𝜓(𝑥)⟹𝑄⁻¹𝑃⁻¹𝑥𝑃𝑄=𝑃⁻¹𝑄⁻¹𝑥𝑄𝑃⟹𝑄𝑃𝑄⁻¹𝑃⁻¹𝑥=𝑥𝑄𝑃𝑄⁻¹𝑃⁻¹,∀𝑥∈𝑠𝑙(𝑛+1,ℂ)_𝑇. (18)
By (3) of the proof, there exists 𝑥^′∈𝑠𝑙(𝑛+1,ℂ) such that 𝑥=𝑇⁻¹𝑥^′. Equation (18) is equivalent to 𝑄𝑃𝑄⁻¹𝑃⁻¹𝑇⁻¹𝑥^′=𝑇⁻¹𝑥^′𝑄𝑃𝑄⁻¹𝑃⁻¹=𝑇⁻¹𝑄𝑃𝑄⁻¹𝑃⁻¹𝑥^′⟹𝑥^′𝑄𝑃𝑄⁻¹𝑃⁻¹=𝑄𝑃𝑄⁻¹𝑃⁻¹𝑥^′. (19)
By the arbitrariness of 𝑥, we have for all 𝑥^′∈𝑠𝑙(𝑛+1,ℂ), (19) is established. According to Schur's lemma we have 𝑄𝑃=𝑘𝑃𝑄, for some 𝑘∈ℂ.
(6) If 𝜙 is an isotopy from (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) to (𝑠𝑙(𝑛+1,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,  𝛼_𝑃); according to Lemma 5, 𝜙 is an isotopy of their Hom-Lie admissible algebras and satisfying 𝛼_𝑃𝜙=𝜙𝛼_𝑄. (20)
By (3) of the theorem, 𝜙 can be defined as 𝜙(𝑥)=𝑇⁻¹𝑥,∀𝑥∈𝑠𝑙(𝑛+1,ℂ). Then ∀𝑥∈𝑠𝑙(𝑛+1,ℂ), (20) is equivalent to 𝛼_𝑃𝜙(𝑥)=𝜙𝛼_𝑄(𝑥)⟹𝑃⁻¹𝑇⁻¹𝑥𝑃=𝑇⁻¹𝑄⁻¹𝑥𝑄⟹𝑇⁻¹𝑃⁻¹𝑥𝑃=𝑇⁻¹𝑄⁻¹𝑥𝑄⟹𝑃⁻¹𝑥𝑃=𝑄⁻¹𝑥𝑄⟹𝑄𝑃⁻¹𝑥=𝑥𝑄𝑃⁻¹. (21) By Schur's lemma, we have 𝑄𝑃⁻¹=𝑘𝐸⇒𝑄=𝑘𝑃, for some 𝑘∈ℂ.
(7) According to (6) of the theorem, 𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_{𝛼𝑘1𝑃},𝛼_𝑘1𝑃≅𝑠𝑙(𝑛+1,ℂ)_𝑇1,[⋅,⋅]_{𝑇1𝛼𝑃},𝛼_𝑃;𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_{𝛼𝑘2𝑄},𝛼_𝑘2𝑄≅𝑠𝑙(𝑛+1,ℂ)_𝑇2,[⋅,⋅]_{𝑇2𝛼𝑄},𝛼_𝑄. (22) Then 𝑠𝑙(𝑛+1,ℂ)_𝑇1,[⋅,⋅]_{𝑇1𝛼𝑃},𝛼_𝑃≅𝑠𝑙(𝑛+1,ℂ)_𝑇2,[⋅,⋅]_{𝑇2𝛼𝑄},𝛼_𝑄⇔𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_{𝛼𝑘1𝑃},𝛼_𝑘1𝑃≅𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_{𝛼𝑘2𝑄},𝛼_𝑘2𝑄. (23) Suppose 𝜙 is an isomorphism of (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_{𝛼𝑘1𝑃},𝛼_𝑘1𝑃) and (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]_{𝛼𝑘2𝑄},𝛼_𝑘2𝑄), and then 𝜙 is an automorphism of (𝑠𝑙(𝑛+1,ℂ),[⋅,⋅]) satisfying 𝛼_𝑘2𝑄𝜙=𝜙𝛼_𝑘1𝑃. (24) By Lie theory 𝜙 can be defined by 𝜙(𝑥)=𝑅⁻¹𝑥𝑅, ∀𝑥∈𝑠𝑙(𝑛+1,ℂ), 𝑅∈𝑔𝐿(𝑉). Then (24) is equivalent to 𝛼_𝑘2𝑄𝜙(𝑥)=𝜙𝛼_𝑘1𝑃(𝑥)⟹𝑘₂𝑄⁻¹𝑅⁻¹𝑥𝑅𝑘₂𝑄=𝑅⁻¹𝑘₁𝑃⁻¹𝑥𝑘₁𝑃𝑅⟹𝑃𝑅𝑄⁻¹𝑅⁻¹𝑥=𝑥𝑃𝑅𝑄⁻¹𝑅⁻¹,∀𝑥∈𝑠𝑙(𝑛+1,ℂ). (25) By Schur's lemma we have 𝑃𝑅𝑄⁻¹𝑅⁻¹=𝑘𝐸⟹𝑃𝑅=𝑘𝑅𝑄⟹𝑃=𝑘𝑅𝑄𝑅⁻¹. (26) That is, 𝑃 and 𝑘𝑄 are conjugate (for some 𝑘∈ℂ).

Theorem 11. Let 𝑉 be a 2𝑛-dimensional vector space with a nondegenerate skew symmetric bilinear form 𝑓∶𝑉×𝑉→ℂ. A subspace 𝑠𝑝(2𝑛,ℂ)_𝑇 of 𝑔𝑙(𝑉) is set 𝑠𝑝2𝑛,ℂ_𝑇=(𝑉)((𝑣))((𝑤=(𝑉)𝑥∈𝑔𝑙∣𝑓𝑥𝑇,𝑤=−𝑓𝑣,𝑥𝑇))𝑥∈𝑔𝑙∣𝑠𝑥𝑇=−𝑇^𝑡𝑥^𝑡𝑠,𝑠=0𝐼_𝑙−𝐼_𝑙0,∀𝑣,𝑤∈𝑉. (27)
(1) On 𝑠𝑝(2𝑛,ℂ)_𝑇, define a bracket [⋅,⋅]_𝑇 by [𝑥,𝑦]_𝑇=𝑥𝑇𝑦−𝑦𝑇𝑥,∀𝑥,𝑦∈𝑠𝑝(2𝑛,ℂ)_𝑇. Then (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇) is a II type Iso-Lie algebra (we call it an 𝐶_𝑛-Iso-Lie algebra). Define an invertible linear map 𝛼_𝑃 by 𝛼_𝑃(𝑥)=𝑃⁻¹𝑥𝑃,∀𝑥∈𝑠𝑝(2𝑛,ℂ)_𝑇, where 𝑃∈𝑔𝐿(𝑉) satisfying 𝑃𝑇=𝑇𝑃;  𝑃^𝑡𝑠𝑃=𝑠. Then 𝛼_𝑃 is an Iso-automorphism of (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇). Furthermore, 𝑆𝑃2𝑛,ℂ_𝑇=𝑃∈𝑔𝐿(𝑉)∣𝑃𝑇=𝑇𝑃,𝑃^𝑡𝑠𝑃=𝑠,𝑠=0𝐼_𝑙−𝐼_𝑙0 (28) is a group (we call it the Iso-symplectic group).
(2) Let 𝑠𝑝(2𝑛,ℂ) be a conventional symplectic Lie algebra. Define an invertible linear map 𝜙 by 𝜙(𝑥)=𝑥𝑇⁻¹,∀𝑥∈𝑠𝑝(2𝑛,ℂ). Then 𝜙 is an isotopy from (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]) to (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇). Furthermore, (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇) is simple.
(3) Let (𝑠𝑝(2𝑛,ℂ)_𝑇1,[⋅,⋅]_𝑇1) and (𝑠𝑝(2𝑛,ℂ)_𝑇2,[⋅,⋅]_𝑇2) be 𝐶_𝑛-Iso-Lie algebras, and then they are isotopic.
(4) On 𝑠𝑝(2𝑛,ℂ)_𝑇, define a new bracket [⋅,⋅]_𝑇𝛼𝑃 by 𝑥,𝑦_𝑇𝛼𝑃=𝛼_𝑃𝑥,𝑦_𝑇=𝑃⁻¹𝑥∀()𝑥𝑇𝑦−𝑦𝑇𝑃,𝑥,𝑦∈𝑠𝑝2𝑛,ℂ_𝑇. (29)
Then (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃) is a simple Iso-Hom-Lie algebra (we call it an 𝐶_𝑛-Iso-Hom-Lie algebra). Define an invertible linear map 𝜓 of 𝑠𝑝(2𝑛,ℂ)_𝑇 by 𝜓(𝑥)=𝑄⁻¹𝑥𝑄,∀𝑥∈𝑠𝑝(2𝑛,ℂ)_𝑇, where 𝑄∈𝑆𝑃(2𝑛,ℂ)_𝑇  satisfies 𝑃𝑄=𝑘𝑄𝑃, for some 𝑘∈ℂ; then 𝜓 is an Iso-automorphism of (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,  𝛼_𝑃).
(5) Let (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) be a Hom-Lie algebra with 𝛼_𝑄(𝑥)=𝑄⁻¹𝑥𝑄;𝑥,𝑦_𝛼𝑄=𝑄⁻¹()().𝑥𝑦−𝑦𝑥𝑄,∀𝑥,𝑦∈𝑠𝑝2𝑛,ℂ,𝑄∈𝑆𝑃2𝑛,ℂ (30)
If 𝑄=±𝑃 is satisfied, then (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) is isotopic to (𝑠𝑝(2𝑛,ℂ)_𝑇,  [⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃).
(6) Let (𝑠𝑝(2𝑛,ℂ)_𝑇1,[⋅,⋅]_{𝑇1𝛼𝑃},𝛼_𝑃) and (𝑠𝑝(2𝑛,ℂ)_𝑇2,[⋅,⋅]_{𝑇2𝛼𝑄},𝛼_𝑄) be 𝐶_𝑛-Iso-Hom-Lie algebras. They are isotopic if and only if 𝑃 and 𝑘𝑄 are conjugate (for some 𝑘∈ℂ).

Proof. (1) Let∀𝑥,𝑦∈𝑠𝑝(2𝑛,ℂ)_𝑇, 𝑠𝑥,𝑦_𝑇𝑇=𝑠𝑥𝑇𝑦−𝑦𝑇𝑥𝑇=−𝑇^𝑡𝑥^𝑡𝑠𝑦𝑇+𝑇^𝑡𝑦^𝑡𝑠𝑥𝑇=𝑇^𝑡𝑥^𝑡𝑇^𝑡𝑦^𝑡𝑠−𝑇^𝑡𝑦^𝑡𝑇^𝑡𝑥^𝑡𝑠=−𝑇^𝑡𝑥,𝑦^𝑡_𝑇𝑠. (31) So [𝑥,𝑦]_𝑇∈𝑠𝑝(2𝑛,ℂ)_𝑇. The Jacobi identity can be checked directly. Hence (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇) is an II-type Iso-Lie algebra.
On 𝑠𝑝(2𝑛,ℂ)_𝑇, define an invertible linear map 𝛼_𝑃 by 𝛼_𝑃(𝑥)=𝑃⁻¹𝑥𝑃,  ∀𝑥∈𝑠𝑝(2𝑛,ℂ)_𝑇, where 𝑃∈𝑔𝐿(𝑉) satisfying 𝑃𝑇=𝑇𝑃;𝑃^𝑡𝑠𝑃=𝑠, we have 𝑠𝑥𝑇=−𝑇^𝑡𝑥^𝑡𝑠⟹𝑃^𝑡𝑠𝑃𝑃⁻¹𝑥𝑃𝑃⁻¹𝑇𝑃=−𝑃^𝑡𝑇^𝑡𝑃^−1𝑡𝑃^𝑡𝑥^𝑡𝑃^−1𝑡𝑃^𝑡𝑠𝑃⟹𝑠𝛼_𝑃(𝑥)𝑇=−𝑇^𝑡𝛼_𝑃(𝑥)^𝑡𝑠. (32) So 𝛼_𝑃(𝑠𝑝(2𝑛,ℂ)_𝑇)⊆𝑠𝑝(2𝑛,ℂ)_𝑇.∀𝑥,𝑦∈𝑠𝑝(2𝑛,ℂ)_𝑇, 𝛼_𝑃𝑥,𝑦_𝑇=𝑃⁻¹𝑥𝑇𝑦−𝑦𝑇𝑥𝑃=𝑃⁻¹𝑥𝑃𝑃⁻¹𝑇𝑃𝑃⁻¹𝑦𝑃−𝑃⁻¹𝑦𝑃𝑃⁻¹𝑇𝑃𝑃⁻¹𝑥𝑃=𝑃⁻¹𝑥𝑃𝑇𝑃⁻¹𝑦𝑃−𝑃⁻¹𝑦𝑃𝑇𝑃⁻¹=𝛼𝑥𝑃_𝑃(𝑥),𝛼_𝑃𝑦_𝑇. (33)
Therefore, 𝛼_𝑃 is an Iso-automorphism of (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇). Let ∀𝑃₁,𝑃₂∈𝑆𝑃(2𝑛,ℂ)_𝑇, 𝑃₁𝑃₂𝑇=𝑃₁𝑇𝑃₂=𝑇𝑃₁𝑃₂;𝑃₁𝑃₂^𝑡𝑠𝑃₁𝑃₂=𝑃^𝑡₂𝑃^𝑡₁𝑠𝑃₁𝑃₂=𝑃^𝑡₂𝑠𝑃₂=𝑠. (34) Hence 𝑃₁𝑃₂∈𝑆𝑃(2𝑛,ℂ)_𝑇. It is obvious that 𝐸 is a unit of 𝑆𝑃(2𝑛,ℂ)_𝑇 and for every element 𝑃∈𝑆𝑃(2𝑛,ℂ)_𝑇, there is an invertible element 𝑃⁻¹∈𝑆𝑃(2𝑛,ℂ)_𝑇. We have that 𝑆𝑃(2𝑛,ℂ)_𝑇 is a group.
(2)   Because 𝑠𝜙(𝑥)𝑇=𝑠𝑥𝑇⁻¹𝑇=𝑠𝑥=−𝑥^𝑡𝑠=−𝑇^𝑡𝜙(𝑥)^𝑡𝑠,∀𝑥∈𝑠𝑝(2𝑛,ℂ), 𝜙𝑠𝑝(2𝑛,ℂ)⊆𝑠𝑝2𝑛,ℂ_𝑇,𝜙𝑥,𝑦=𝜙(𝑥),𝜙𝑦_𝑇,∀𝑥,𝑦∈𝑠𝑝(2𝑛,ℂ) (35) can be checked as in (3) of Theorem 10. Therefore, 𝜙 is an isotopy from (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]) to (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇). (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇) is simple and can be proved as (3) of Theorem 10.
(3)  Define an invertible linear map 𝜙 by 𝜙(𝑥)=𝑥𝑇₁𝑇₂⁻¹,∀𝑥∈𝑠𝑝(2𝑛,ℂ)_𝑇1. Then 𝜙(𝑥)∈𝑠𝑝2𝑛,ℂ_𝑇2,𝜙𝑥,𝑦_𝑇1=𝜙(𝑥),𝜙𝑦_𝑇2,∀𝑥,𝑦∈𝑠𝑝2𝑛,ℂ_𝑇1 (36) can be proved as (2) of the proof. Therefore, 𝜙 is an isotopy from (𝑠𝑝(2𝑛,ℂ)_𝑇1,  [⋅,⋅]_𝑇1) to (𝑠𝑝(2𝑛,ℂ)_𝑇2,[⋅,⋅]_𝑇2).
(4) (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃) is a simple Iso-Hom-Lie algebra and can be got from (1) and (3) of Theorem, Lemma 3, and Proposition 7 directly. Suppose 𝜓 is an Iso-automorphism of (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃), and according to Lemma 6, 𝜓 is an Iso-automorphism of the Hom-Lie admissible algebra (𝑠𝑝(2𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇) satisfying 𝛼_𝑃𝜓=𝜓𝛼_𝑃. (37) By (1) of the theorem, 𝜓 can be defined by 𝜓(𝑥)=𝑄⁻¹𝑥𝑄,∀𝑥∈𝑠𝑝(2𝑛,ℂ)_𝑇, where 𝑄∈𝑆𝑃(2𝑛,ℂ)_𝑇. So (37) is equivalent to 𝛼_𝑃𝜓(𝑥)=𝜓𝛼_𝑃(𝑥)=𝑃⁻¹𝑄⁻¹𝑥𝑄𝑃=𝑄⁻¹𝑃⁻¹𝑥𝑃𝑄⟹𝑃𝑄𝑃⁻¹𝑄⁻¹𝑥=𝑥𝑃𝑄𝑃⁻¹𝑄⁻¹,∀𝑥∈𝑠𝑝2𝑛,ℂ_𝑇. (38) By (2) of the proof, ∃𝑥^′∈𝑠𝑝(2𝑛,ℂ) such that 𝑥=𝑥^′𝑇⁻¹. Then (38) is equivalent to 𝑃𝑄𝑃⁻¹𝑄⁻¹𝑥^′𝑇⁻¹=𝑥^′𝑇⁻¹𝑃𝑄𝑃⁻¹𝑄⁻¹=𝑥^′𝑃𝑄𝑃⁻¹𝑄⁻¹𝑇⁻¹⟹𝑃𝑄𝑃⁻¹𝑄⁻¹𝑥^′=𝑥^′𝑃𝑄𝑃⁻¹𝑄⁻¹. (39) By the arbitrariness of 𝑥, we have for all 𝑥^′∈𝑠𝑝(2𝑛,ℂ), (39) is established. According to Schur's lemma, we have 𝑃𝑄=𝑘𝑄𝑃, for some 𝑘∈ℂ.
(5) Suppose 𝜙 is an isotopy from (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) to (𝑠𝑝(2𝑛,ℂ)_𝑇,  [⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃). According to Lemma 5, 𝜙 is an isotopy of their Hom-Lie admissible algebras satisfying 𝛼_𝑃𝜙=𝜙𝛼_𝑄. (40) By (2) of theorem, 𝜙 can be defined by 𝜙(𝑥)=𝑥𝑇⁻¹,∀𝑥∈𝑠𝑝(2𝑛,ℂ). Equation(40) is equivalent to 𝛼_𝑃𝜙(𝑥)=𝜙𝛼_𝑄(𝑥)⟹𝑃⁻¹𝑥𝑇⁻¹𝑃=𝑄⁻¹𝑥𝑄𝑇⁻¹⟹𝑃⁻¹𝑥𝑃𝑇⁻¹=𝑄⁻¹𝑥𝑄𝑇⁻¹⟹𝑃⁻¹𝑥𝑃=𝑄⁻¹𝑥𝑄⟹𝑄𝑃⁻¹𝑥=𝑥𝑄𝑃⁻¹,∀𝑥∈𝑠𝑝(2𝑛,ℂ). (41) By Schur's lemma we have 𝑄=𝑘𝑃, for some 𝑘∈ℂ. According to (4) of the theorem, 𝑃∈𝑆𝑝(2𝑛,ℂ)_𝑇, so 𝑃^𝑡𝑠𝑃=𝑠. Because 𝑄∈𝑆𝑃(2𝑛,ℂ), then 𝑄^𝑡𝑠𝑄=𝑠⇒𝑘𝑃^𝑡𝑠𝑘𝑃=𝑘²𝑃^𝑡𝑠𝑃=𝑘²𝑠=𝑠⇒𝑘=±1. We have the conclusion.
(6) The same reason as (7) of Theorem 10, (𝑠𝑝(2𝑛,ℂ)_𝑇1,[⋅,⋅]_{𝑇1𝛼𝑃},  𝛼_𝑃) and (𝑠𝑝(2𝑛,ℂ)_𝑇2,[⋅,⋅]_{𝑇2𝛼𝑄},𝛼_𝑄) are isotopic if and only if (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]_𝛼±𝑃,𝛼_±𝑃)≅(𝑠𝑝(2𝑛,ℂ),[⋅,⋅]_𝛼±𝑄,𝛼_±𝑄). Suppose 𝜓 is an isomorphism from (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]_𝛼±𝑃,𝛼_±𝑃) to (𝑠𝑝(2𝑛,ℂ),[⋅,⋅]_𝛼±𝑄,𝛼_±𝑄), then 𝜓 is an automorphism of (𝑠𝑝(2𝑛,ℂ),[⋅,⋅])   satisfying 𝛼_±𝑄𝜓=𝜓𝛼_±𝑃. (42) By Lie theory we know 𝜓 can be defined by 𝜓(𝑥)=𝑅⁻¹𝑥𝑅,∀𝑥∈𝑠𝑝(2𝑛,ℂ), where  𝑅∈𝑆𝑃(2𝑛,ℂ) and (42) is equivalent to 𝛼_±𝑄𝜓(𝑥)=𝜓𝛼_±𝑃(𝑥)⟹(±𝑄)⁻¹𝑅⁻¹𝑥𝑅(±𝑄)=𝑅⁻¹(±𝑃)⁻¹𝑥(±𝑃)𝑅⟹𝑃𝑅𝑄⁻¹𝑅⁻¹𝑥=𝑥𝑃𝑅𝑄⁻¹𝑅⁻¹,∀𝑥∈𝑠𝑝(2𝑛,ℂ). (43) By Schur's lemma we have 𝑃𝑅𝑄⁻¹𝑅⁻¹=𝑘𝐸⇒𝑅⁻¹𝑃𝑅=𝑘𝑄. That is, 𝑃 is conjugate with 𝑘𝑄 (for some 𝑘∈ℂ).

Theorem 12. Let 𝑉 be an 𝑛-dimensional vector space with a nondegenerate symmetric bilinear form 𝑓∶𝑉×𝑉→ℂ. Define a subspace 𝑠𝑜(𝑛,ℂ)_𝑇 of 𝑔𝑙(𝑉) by 𝑠𝑜𝑛,ℂ_𝑇=𝑥∈𝑔𝑙(𝑉)∣𝑓(𝑥𝑇(𝑣),𝑤)=−𝑓(𝑣,𝑥𝑇(𝑤))=𝑥∈𝑔𝑙(𝑉)∣𝑠𝑥𝑇=−𝑇^𝑡𝑥^𝑡𝑠,𝑠=10000𝐼_𝑙0𝐼_𝑙0,𝑛=2𝑙+10𝐼_𝑙𝐼_𝑙0.,𝑛=2𝑙∀𝑣,𝑤∈𝑉 (44)
(1) On 𝑠𝑜(𝑛,ℂ)_𝑇 define a bracket [⋅,⋅]_𝑇 by [𝑥,𝑦]_𝑇=𝑥𝑇𝑦−𝑦𝑇𝑥,∀𝑥,𝑦∈𝑠𝑜(𝑛,ℂ)_𝑇, and then (𝑠𝑜(𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇) is an II type𝐵_𝑙(𝑛=2𝑙+1) or 𝐷_𝑙(𝑛=2𝑙) Iso-Lie algebra.
Define an invertible linear map 𝛼_𝑃 by 𝛼_𝑃(𝑥)=𝑃⁻¹𝑥𝑃,∀𝑥∈𝑠𝑜(𝑛,ℂ)_𝑇, where 𝑃∈𝑔𝐿(𝑉) satisfying 𝑃𝑇=𝑇𝑃,𝑃^𝑡𝑠𝑃=𝑠. Then 𝛼_𝑃 is an Iso-automorphism of (𝑠𝑜(𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇). Furthermore, 𝑆𝑂𝑛,ℂ_𝑇=𝑃∈𝑔𝐿(𝑉)∣𝑃𝑇=𝑇𝑃,𝑃^𝑡𝑠𝑃=𝑠,𝑠=10000𝐼_𝑙0𝐼_𝑙0,𝑛=2𝑙+10𝐼_𝑙𝐼_𝑙0,𝑛=2𝑙 (45)is a group (we call it the Iso-orthogonal group).
(2) Let (𝑠𝑜(𝑛,ℂ),[⋅,⋅]) be a conventional orthogonal Lie algebra. On 𝑠𝑜(𝑛,ℂ), define an invertible linear map 𝜙 by 𝜙(𝑥)=𝑥𝑇⁻¹,∀𝑥∈𝑠𝑜(𝑛,ℂ). Then 𝜙 is an isotopy from the conventional Lie algebra (𝑠𝑜(𝑛,ℂ),[⋅,⋅]) to (𝑠𝑜(𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇). Furthermore (𝑠𝑜(𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇) is simple.
(3) Let (𝑠𝑜(𝑛,ℂ)_𝑇1,[⋅,⋅]_𝑇1) and (𝑠𝑜(𝑛,ℂ)_𝑇2,[⋅,⋅]_𝑇2) be 𝐵_𝑙 (or 𝐷_𝑙)-Iso-Lie algebras, and then they are isotopic.
(4) On 𝑠𝑜(𝑛,ℂ)_𝑇, define a new bracket [⋅,⋅]_𝑇𝛼𝑃 by 𝑥,𝑦_𝑇𝛼𝑃=𝛼_𝑃𝑥,𝑦_𝑇=𝑃⁻¹𝑥𝑇𝑦−𝑦𝑇𝑥𝑃,∀𝑥,𝑦∈𝑠𝑜𝑛,ℂ_𝑇. (46)
Then (𝑠𝑜(𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃) is a simple Iso-Hom-Lie algebra (we call it a 𝐵_𝑙 or 𝐷_𝑙-Iso-Hom-Lie algebra). On 𝑠𝑜(𝑛,ℂ)_𝑇, define an invertible linear map 𝜓 by 𝜓(𝑥)=𝑄⁻¹𝑥𝑄,∀𝑥∈𝑠𝑜(𝑛,ℂ)_𝑇, where 𝑄∈𝑆𝑂(𝑛,ℂ)_𝑇 satisfies 𝑃𝑄=𝑘𝑄𝑃, for some 𝑘∈ℂ, and then 𝜓 is an Iso-automorphism of (𝑠𝑜(𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃).
(5) Let (𝑠𝑜(𝑛,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) be a Hom-Lie algebra with 𝛼_𝑄(𝑥)=𝑄⁻¹𝑥𝑄;𝑥,𝑦_𝛼𝑄=𝑄⁻¹()().𝑥𝑦−𝑦𝑥𝑄,∀𝑥,𝑦∈𝑠𝑜𝑛,ℂ,𝑄∈𝑆𝑂𝑛,ℂ (47)
If 𝑄=±𝑃 is satisfied, then (𝑠𝑜(𝑛,ℂ),[⋅,⋅]_𝛼𝑄,𝛼_𝑄) is isotopic to (𝑠𝑜(𝑛,ℂ)_𝑇,[⋅,⋅]_𝑇𝛼𝑃,𝛼_𝑃).
(6) Let (𝑠𝑜(𝑛,ℂ)_𝑇1,[⋅,⋅]_{𝑇1𝛼𝑃},𝛼_𝑃) and (𝑠𝑜(𝑛,ℂ)_𝑇2,[⋅,⋅]_{𝑇2𝛼𝑄},𝛼_𝑄) be 𝐵_𝑙 (or 𝐷_𝑙)-Iso-Hom-Lie algebras. They are isotopic if and only if 𝑃 and 𝑘𝑄 are conjugate (for some 𝑘∈ℂ).

Acknowledgment

This project is partially supported by the National Nature science foundation of China (11071187).