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

Authors: Xiuxian Li
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 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 I^ 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 overℂ 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 Section2, 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 “xt” 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
(1)[x,y]=-[y,x],∀x,y∈𝔤;[α(x),[y,z]]+[α(y),[z,x]]+[α(z),[x,y]]=0,∀x,y,z∈𝔤.Whenα=id, (𝔤,[·,·],α) is degenerated to Lie algebra (𝔤,[·,·]). A Hom-Lie algebra (𝔤,[·,·],α) is called multiplicative if
(2)α([x,y])=[α(x),α(y)],∀x,y∈𝔤
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 subspace𝔤1 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 automorphismψ 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 thatKer(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
(3)α-1([x,y]′)=[α-1(x),α-1(y)]′,∀x,y∈𝔤.
Now we prove that (𝔤,[·,·]') is a Lie algebra. The skew-symmetricity of [·,·]′ is obvious. ∀x,y,z∈𝔤,
(4)↺x,y,z[x,[y,z]′]′=↺x,y,zα-1([x,α-1([y,z])])=↺x,y,zα-1([α-1α(x),α-1([y,z])])=↺x,y,zα-2([α(x),[y,z]])=0,
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 Theorem4, 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
(5)[𝔤,𝔤]′=α-1([𝔤,𝔤])=[α-1(𝔤),α-1(𝔤)]⊆[𝔤,𝔤]⫋𝔤,
which is a contradiction with the simplicity of (𝔤,[·,·]'). This reduces to [𝔤,𝔤]=𝔤.
Let𝔤1 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

Let𝔤 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 IIAn,  Bl,  Cn, Dl-Iso-(Hom-)Lie algebras over ℂ. Give their Iso-automorphisms and isotopies. Prove that they are simple.LetV 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, 𝔤T⊆gl(V). (𝔤T,[·,·]T) is an II type Iso-Lie algebra over ℂ, with the bracket defined by [x,y]T=xTy-yTx,∀x,y∈𝔤T. 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.
Letsl(n+1,ℂ^)T be a vector space satisfying
(6)sl(n+1,ℂ^)T={x^∈gl(V)∣Tr(Tx^)=0}.(1)
Define a bracket[·,·]T by [x^,y^]T=x^Ty^-y^Tx^,∀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 An-Iso-Lie algebra).(2)
Define an invertible linear mapαP by
(7)αP(x^)=P-1x^P,∀x^∈sl(n+1,ℂ)T^,whereP∈gL(V) satisfying PT=TP, and then αP is an Iso-automorphism of (sl(n+1,ℂ^)T,[·,·]T).(3)
There is an isotopy between the conventionalAn-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 An-Iso-Lie algebras, and then they are isotopic.(5)
Define a new bracket[·,·]TαP by
(8)([x^,y^]TαP)=αP([x^,y^]T)=P-1(x^Ty^-y^Tx^)P,∀x^,y^∈sl(n+1,ℂ^)T.Then(sl(n+1,ℂ^)T,[·,·]TαP,αP) is a simple Iso-Hom-Lie algebra (one calls it an An-Iso-Hom-Lie algebra) with an Iso-automorphism ψ defined by ψ(x^)=Q-1x^Q,∀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,αQ) be a Hom-Lie algebra with
(9)αQ(x)=Q-1xQ;[x,y]αQ=Q-1(xy-yx)Q,∀x,y∈sl(n+1,ℂ).IfQ=kP, for some k∈ℂ is satisfied, then (sl(n+1,ℂ),[·,·]αQ,αQ) is isotopic to (sl(n+1,ℂ^)T,[·,·]TαP,αP).(7)
Let(sl(n+1,ℂ^)T1,[·,·]T1αP,αP) and (sl(n+1,ℂ^)T2,[·,·]T2αQ,αQ) be An-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(Tx^Ty^-Ty^Tx^)=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) BecauseTP=PT, so TP-1=P-1T and P-1TP=T are satisfied. Consider ∀x^∈sl(n+1,ℂ^)T,
(10)Tr(TαP(x^))=Tr(TP-1x^P)=Tr(P-1Tx^P)=Tr(Tx^)=0.
Therefore αP(sl(n+1,ℂ^)T)⊆sl(n+1,ℂ^)T.  ∀x^,y^∈sl(n+1,ℂ^)T,
(11)αP([x^,y^]T)=P-1(x^Ty^-y^Tx^)P=P-1x^PP-1TPP-1y^P-P-1y^PP-1TPP-1x^P=P-1x^PTP-1y^P-P-1y^PTP-1x^P=[αP(x^),αP(y^)]T.
So αP is an Iso-automorphism of (sl(n+1,ℂ^)T,[·,·]T).
(3) Define an invertible linear mapϕ by
(12)ϕ(x)=T-1x,∀x∈sl(n+1,ℂ).
It is obvious that ϕ is invertible and Tr(Tϕ(x))=0; therefore, ϕ(sl(n+1,ℂ))⊆sl(n+1,ℂ^)T.
(13)[ϕ(x),ϕ(y)]T=ϕ(x)Tϕ(y)-ϕ(y)Tϕ(x)=T-1xTT-1y-T-1yTT-1x=T-1(xy-yx)=ϕ([x,y]),∀x,y∈sl(n+1,ℂ).
We have the first conclusion.
Suppose𝔤^T is a nontrivial ideal of (sl(n+1,ℂ^)T,[·,·]T). Let 𝔤=T𝔤^T=ϕ-1(𝔤^T). Then
(14)[𝔤,sl(n+1,ℂ)]=ϕ-1[ϕ(𝔤),ϕ(sl(n+1,ℂ))]⊆ϕ-1([𝔤^T,sl(n+1,ℂ^)T]T)⊆ϕ-1(𝔤^T)=𝔤.
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 mapϕ by
(15)ϕ(x^)=x^T1T2-1,∀x^∈sl(n+1,ℂ^)T1;
then
(16)ϕ(x^)∈sl(n+1,ℂ^)T2,ϕ([x^,y^]T1)=[ϕ(x^),ϕ(y^)]T2,∀x^,y^∈sl(n+1,ℂ^)T1
can be checked as in (3) of the theorem. We have the conclusion.
(5) According to (2) and (3) of the theorem, Lemma3, Proposition 7, and the definition of Iso-Hom-Lie algebra, we have that (sl(n+1,ℂ^)T,[·,·]TαP,αP) is an simple Iso-Hom-Lie algebra. If ψ is an Iso-automorphism of (sl(n+1,ℂ^)T,  [·,·]TαP,αP), by Lemma 6, ψ is an Iso-automorphism of the Hom-Lie admissible algebra (sl(n+1,ℂ^)T,[·,·]T) and satisfying
(17)ψαP=αPψ.
By (2) of the theorem again we haveψ(x^)=Q-1x^Q,∀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
(18)ψαP(x^)=αPψ(x^)⟹Q-1P-1x^PQ=P-1Q-1x^QP⟹QPQ-1P-1x^=x^QPQ-1P-1,∀x^∈sl(n+1,ℂ)T^.
By (3) of the proof, there existsx′∈sl(n+1,ℂ) such that x^=T-1x′. Equation (18) is equivalent to
(19)QPQ-1P-1T-1x′=T-1x′QPQ-1P-1=T-1QPQ-1P-1x′⟹x′QPQ-1P-1=QPQ-1P-1x′.
By the arbitrariness ofx^, 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) Ifϕ is an isotopy from (sl(n+1,ℂ),[·,·]αQ,αQ) to (sl(n+1,ℂ)T^,[·,·]TαP,  αP); according to Lemma 5, ϕ is an isotopy of their Hom-Lie admissible algebras and satisfying
(20)αPϕ=ϕαQ.
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
(21)αPϕ(x)=ϕαQ(x)⟹P-1T-1xP=T-1Q-1xQ⟹T-1P-1xP=T-1Q-1xQ⟹P-1xP=Q-1xQ⟹QP-1x=xQP-1.
By Schur's lemma, we have QP-1=kE⇒Q=kP, for some k∈ℂ.
(7) According to (6) of the theorem,(22)(sl(n+1,ℂ),[·,·]αk1P,αk1P)≅(sl(n+1,ℂ)T1^,[·,·]T1αP,αP);(sl(n+1,ℂ),[·,·]αk2Q,αk2Q)≅(sl(n+1,ℂ)T2^,[·,·]T2αQ,αQ).
Then
(23)(sl(n+1,ℂ)T1^,[·,·]T1αP,αP)≅(sl(n+1,ℂ)T2^,[·,·]T2αQ,αQ)⇔(sl(n+1,ℂ),[·,·]αk1P,αk1P)≅(sl(n+1,ℂ),[·,·]αk2Q,αk2Q).
Suppose ϕ is an isomorphism of (sl(n+1,ℂ),[·,·]αk1P,αk1P) and (sl(n+1,ℂ),[·,·]αk2Q,αk2Q), and then ϕ is an automorphism of (sl(n+1,ℂ),[·,·]) satisfying
(24)αk2Qϕ=ϕαk1P.
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
(25)αk2Qϕ(x)=ϕαk1P(x)⟹(k2Q)-1R-1xR(k2Q)=R-1(k1P)-1x(k1P)R⟹PRQ-1R-1x=xPRQ-1R-1,∀x∈sl(n+1,ℂ).
By Schur's lemma we have
(26)PRQ-1R-1=kE⟹PR=kRQ⟹P=kRQR-1.
That is, P and kQ are conjugate (for some k∈ℂ).Theorem 11.
LetV 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
(27)sp(2n,ℂ^)T={x^∈gl(V)∣f(x^T(v),w)=-f(v,x^T(w))}={x^∈gl(V)∣sx^T=-Ttx^ts,s=(0Il-Il0)},∀v,w∈V.
(1) Onsp(2n,ℂ^)T, define a bracket [·,·]T by [x^,y^]T=x^Ty^-y^Tx^,∀x^,y^∈sp(2n,ℂ^)T. Then (sp(2n,ℂ)T,[·,·]T) is a II type Iso-Lie algebra (we call it an Cn-Iso-Lie algebra). Define an invertible linear map αP by αP(x^)=P-1x^P,∀x^∈sp(2n,ℂ^)T, where P∈gL(V) satisfying PT=TP;  PtsP=s. Then αP is an Iso-automorphism of (sp(2n,ℂ^)T,[·,·]T). Furthermore,
(28)SP(2n,ℂ^)T={P∈gL(V)∣PT=TP,PtsP=s,s=(0Il-Il0)}
is a group (we call it the Iso-symplectic group).
(2) Letsp(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 Cn-Iso-Lie algebras, and then they are isotopic.
(4) Onsp(2n,ℂ^)T, define a new bracket [·,·]TαP by
(29)([x^,y^]TαP)=αP([x^,y^]T)=P-1(x^Ty^-y^Tx^)P,∀x^,y^∈sp(2n,ℂ)T^.
Then(sp(2n,ℂ^)T,[·,·]TαP,αP) is a simple Iso-Hom-Lie algebra (we call it an Cn-Iso-Hom-Lie algebra). Define an invertible linear map ψ of sp(2n,ℂ^)T by ψ(x^)=Q-1x^Q,∀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,  αP).
(5) Let(sp(2n,ℂ),[·,·]αQ,αQ) be a Hom-Lie algebra with
(30)αQ(x)=Q-1xQ;[x,y]αQ=Q-1(xy-yx)Q,∀x,y∈sp(2n,ℂ),Q∈SP(2n,ℂ).
IfQ=±P is satisfied, then (sp(2n,ℂ),[·,·]αQ,αQ) is isotopic to (sp(2n,ℂ^)T,  [·,·]TαP,αP).
(6) Let(sp(2n,ℂ^)T1,[·,·]T1αP,αP) and (sp(2n,ℂ^)T2,[·,·]T2αQ,αQ) be Cn-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,
(31)s[x^,y^]TT=s(x^Ty^-y^Tx^)T=-Ttx^tsy^T+Tty^tsx^T=Ttx^tTty^ts-Tty^tTtx^ts=-Tt[x^,y^]Tts.
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.
Onsp(2n,ℂ^)T, define an invertible linear map αP by αP(x^)=P-1x^P,  ∀x^∈sp(2n,ℂ^)T, where P∈gL(V) satisfying PT=TP;PtsP=s, we have
(32)sx^T=-Ttx^ts⟹PtsPP-1x^PP-1TP=-PtTt(P-1)tPtx^t(P-1)tPtsP⟹sαP(x^)T=-TtαP((x^)t)s.
So αP(sp(2n,ℂ^)T)⊆sp(2n,ℂ^)T.∀x^,y^∈sp(2n,ℂ^)T,
(33)αP([x^,y^]T)=P-1(x^Ty^-y^Tx^)P=P-1x^PP-1TPP-1y^P-P-1y^PP-1TPP-1x^P=P-1x^PTP-1y^P-P-1y^PTP-1x^P=[αP(x^),αP(y^)]T.
Therefore,αP is an Iso-automorphism of (sp(2n,ℂ^)T,[·,·]T).  Let ∀P1,P2∈SP(2n,ℂ^)T,
(34)P1P2T=P1TP2=TP1P2;(P1P2)ts(P1P2)=P2tP1tsP1P2=P2tsP2=s.
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)   Becausesϕ(x)T=sxT-1T=sx=-xts=-Ttϕ(x)ts,∀x∈sp(2n,ℂ),
(35)ϕ(sp(2n,ℂ))⊆sp(2n,ℂ^)T,ϕ([x,y])=[ϕ(x),ϕ(y)]T,∀x,y∈sp(2n,ℂ)
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 mapϕ by ϕ(x^)=x^T1T2-1,∀x^∈sp(2n,ℂ^)T1. Then
(36)ϕ(x^)∈sp(2n,ℂ^)T2,ϕ([x^,y^]T1)=[ϕ(x^),ϕ(y^)]T2,∀x^,y^∈sp(2n,ℂ^)T1
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,αP) 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,αP), and according to Lemma 6, ψ is an Iso-automorphism of the Hom-Lie admissible algebra (sp(2n,ℂ^)T,[·,·]T) satisfying
(37)αPψ=ψαP.
By (1) of the theorem, ψ can be defined by ψ(x^)=Q-1x^Q,∀x^∈sp(2n,ℂ^)T, where Q∈SP(2n,ℂ^)T. So (37) is equivalent to
(38)αPψ(x^)=ψαP(x^)=P-1Q-1x^QP=Q-1P-1x^PQ⟹PQP-1Q-1x^=x^PQP-1Q-1,∀x^∈sp(2n,ℂ^)T.
By (2) of the proof, ∃x′∈sp(2n,ℂ) such that x^=x′T-1. Then (38) is equivalent to
(39)PQP-1Q-1x′T-1=x′T-1PQP-1Q-1=x′PQP-1Q-1T-1⟹PQP-1Q-1x′=x′PQP-1Q-1.
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) Supposeϕ is an isotopy from (sp(2n,ℂ),[·,·]αQ,αQ) to (sp(2n,ℂ^)T,  [·,·]TαP,αP). According to Lemma 5, ϕ is an isotopy of their Hom-Lie admissible algebras satisfying
(40)αPϕ=ϕαQ.
By (2) of theorem, ϕ can be defined by ϕ(x)=xT-1,∀x∈sp(2n,ℂ). Equation(40) is equivalent to
(41)αPϕ(x)=ϕαQ(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).
By Schur's lemma we have Q=kP, for some k∈ℂ. According to (4) of the theorem, P∈Sp(2n,ℂ^)T, so PtsP=s. Because Q∈SP(2n,ℂ), then QtsQ=s⇒kPtskP=k2PtsP=k2s=s⇒k=±1. We have the conclusion.
(6) The same reason as (7) of Theorem10, (sp(2n,ℂ^)T1,[·,·]T1αP,  αP) and (sp(2n,ℂ^)T2,[·,·]T2αQ,αQ) 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
(42)α±Qψ=ψα±P.
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
(43)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).
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.
LetV 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 (44)so(n,ℂ^)T={x^∈gl(V)∣f(x^T(v),w)=-f(v,x^T(w))}={x^∈gl(V)∣sx^T=-Ttx^ts,s={(10000Il0Il0),n=2l+1(0IlIl0),n=2l}.∀v,w∈V
(1) Onso(n,ℂ^)T define a bracket [·,·]T by [x^,y^]T=x^Ty^-y^Tx^,∀x^,y^∈so(n,ℂ^)T, and then (so(n,ℂ^)T,[·,·]T) is an II typeBl(n=2l+1) or Dl(n=2l) Iso-Lie algebra.
Define an invertible linear mapαP by αP(x^)=P-1x^P,∀x^∈so(n,ℂ^)T, where P∈gL(V) satisfying PT=TP,PtsP=s. Then αP is an Iso-automorphism of (so(n,ℂ^)T,[·,·]T). Furthermore, (45)SO(n,ℂ^)T={P∈gL(V)∣PT=TP,PtsP=s,s={(10000Il0Il0),n=2l+1(0IlIl0),n=2l}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 Bl (or Dl)-Iso-Lie algebras, and then they are isotopic.
(4) Onso(n,ℂ^)T, define a new bracket [·,·]TαP by
(46)([x^,y^]TαP)=αP([x^,y^]T)=P-1(x^Ty^-y^Tx^)P,∀x^,y^∈so(n,ℂ^)T.
Then(so(n,ℂ^)T,[·,·]TαP,αP) is a simple Iso-Hom-Lie algebra (we call it a Bl or Dl-Iso-Hom-Lie algebra). On so(n,ℂ^)T, define an invertible linear map ψ by ψ(x^)=Q-1x^Q,∀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,αP).
(5) Let(so(n,ℂ),[·,·]αQ,αQ) be a Hom-Lie algebra with
(47)αQ(x)=Q-1xQ;[x,y]αQ=Q-1(xy-yx)Q,∀x,y∈so(n,ℂ),Q∈SO(n,ℂ).
IfQ=±P is satisfied, then (so(n,ℂ),[·,·]αQ,αQ) is isotopic to (so(n,ℂ^)T,[·,·]TαP,αP).
(6) Let(so(n,ℂ^)T1,[·,·]T1αP,αP) and (so(n,ℂ^)T2,[·,·]T2αQ,αQ) be Bl (or Dl)-Iso-Hom-Lie algebras. They are isotopic if and only if P and kQ are conjugate (for some k∈ℂ).



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*Source: 275250-2013-07-29.xml*