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 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

()

When α = id, (𝔤, [·, ·], α) is degenerated to Lie algebra (𝔤, [·, ·]). A Hom-Lie algebra (𝔤, [·, ·], α) is called multiplicative if

()

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

()

Now we prove that (𝔤, [·, ·]′) is a Lie algebra. The skew-symmetricity of [·, ·] ^′ is obvious. ∀x, y, z ∈ 𝔤,

()

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

()

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

, specially, in this paper we study Iso-algebra over ℂ):

I: x*y = axy, 0 ≠ a ∈ ℂ is fixed;
II: x*y = xTy, ∀x, y ∈ 𝔤, T is 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 be a vector space satisfying

()

(1)
Define a bracket [·, ·] _T by . Then 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
()
where P ∈ gL(V) satisfying PT = TP, and then α_P is an Iso-automorphism of .
(3)
There is an isotopy between the conventional A_n-type Lie algebra (sl(n + 1, ℂ), [·, ·]) and . Moreover, is simple.
(4)
Let and be A_n-Iso-Lie algebras, and then they are isotopic.
(5)
Define a new bracket by
()
Then is a simple Iso-Hom-Lie algebra (one calls it an A_n-Iso-Hom-Lie algebra) with an Iso-automorphism ψ defined by , where Q ∈ gL(V) satisfying TQ = QT and PQ = kQP, for some k ∈ ℂ.
(6)
Let be a Hom-Lie algebra with
()
If Q = kP, for some k ∈ ℂ is satisfied, then is isotopic to .
(7)
Let and 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) , there are ; , then ; therefore, . The Jacobi identity can be checked directly. By Definition 8, , [·, ·] _T) is an II-type Iso-Lie algebra.

(2) Because TP = PT, so TP⁻¹ = P⁻¹T and P⁻¹TP = T are satisfied. Consider ,

()

Therefore

()

So α_P is an Iso-automorphism of

(3) Define an invertible linear map ϕ by

()

It is obvious that ϕ is invertible and Tr(Tϕ(x)) = 0; therefore,

()

We have the first conclusion.

Suppose is a nontrivial ideal of . Let . Then

()

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

is simple.

(4) Define an invertible linear map ϕ by

()

then

()

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 is an simple Iso-Hom-Lie algebra. If ψ is an Iso-automorphism of , , by Lemma 6, ψ is an Iso-automorphism of the Hom-Lie admissible algebra and satisfying

()

By (2) of the theorem again we have , where Q ∈ gL(V) satisfies QT = TQ. Consider , and (17) is equivalent to

()

By (3) of the proof, there exists x^′ ∈ sl(n + 1, ℂ) such that . Equation (18) is equivalent to

()

By the arbitrariness of , 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 to , α_P); according to Lemma 5, ϕ is an isotopy of their Hom-Lie admissible algebras and satisfying

()

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

()

By Schur′s lemma, we have QP⁻¹ = kE⇒Q = kP, for some k ∈ ℂ.

(7) According to (6) of the theorem,

()

Then

()

Suppose ϕ is an isomorphism of

and

, and then ϕ is an automorphism of (sl(n + 1, ℂ), [·, ·]) satisfying

()

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

()

By Schur′s lemma we have

()

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 of gl(V) is set

()

(1) On , define a bracket [·, ·] _T by . 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 , where P ∈ gL(V) satisfying PT = TP; P^tsP = s. Then α_P is an Iso-automorphism of . Furthermore,

()

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 . Furthermore, is simple.

(3) Let and be C_n-Iso-Lie algebras, and then they are isotopic.

(4) On , define a new bracket by

()

Then is a simple Iso-Hom-Lie algebra (we call it an C_n-Iso-Hom-Lie algebra). Define an invertible linear map ψ of by , where satisfies PQ = kQP, for some k ∈ ℂ; then ψ is an Iso-automorphism of , α_P).

(5) Let be a Hom-Lie algebra with

()

If Q = ±P is satisfied, then is isotopic to , .

(6) Let and 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,

()

. The Jacobi identity can be checked directly. Hence

is an II-type Iso-Lie algebra.

On , define an invertible linear map α_P by , , where P ∈ gL(V) satisfying PT = TP; P^tsP = s, we have

()

Therefore, α_P is an Iso-automorphism of . Let ,

()

Hence

. It is obvious that E is a unit of

and for every element

, there is an invertible element

. We have that

is a group.

(2) Because sϕ(x)T = sxT⁻¹T = sx = −x^ts = −T^tϕ(x) ^ts, ∀ x ∈ sp(2n, ℂ),

()

can be checked as in (3) of Theorem 10. Therefore, ϕ is an isotopy from (sp(2n, ℂ), [·, ·]) to

is simple and can be proved as (3) of Theorem 10.

(3) Define an invertible linear map ϕ by . Then

()

can be proved as (2) of the proof. Therefore, ϕ is an isotopy from

(4) 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 , and according to Lemma 6, ψ is an Iso-automorphism of the Hom-Lie admissible algebra satisfying

()

By (1) of the theorem, ψ can be defined by

, where

. So (37) is equivalent to

()

By (2) of the proof, ∃x^′ ∈ sp(2n, ℂ) such that

. Then (38) is equivalent to

()

By the arbitrariness of

, 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 to , . According to Lemma 5, ϕ is an isotopy of their Hom-Lie admissible algebras satisfying

()

By (2) of theorem, ϕ can be defined by ϕ(x) = xT⁻¹, ∀ x ∈ sp(2n, ℂ). Equation(40) is equivalent to

()

By Schur′s lemma we have Q = kP, for some k ∈ ℂ. According to (4) of the theorem,

, 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, , α_P) and are isotopic if and only if . Suppose ψ is an isomorphism from to , then ψ is an automorphism of (sp(2n, ℂ), [·, ·]) satisfying

()

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

()

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 of gl(V) by

()

(1) On define a bracket [·, ·] _T by , and then is an II type B_l(n = 2l + 1) or D_l (n = 2l) Iso-Lie algebra.

Define an invertible linear map α_P by , where P ∈ gL(V) satisfying PT = TP, P^tsP = s. Then α_P is an Iso-automorphism of . Furthermore,

()

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 . Furthermore is simple.

(3) Let and be B_l (or D_l)-Iso-Lie algebras, and then they are isotopic.

(4) On , define a new bracket by

()

Then is a simple Iso-Hom-Lie algebra (we call it a B_l or D_l-Iso-Hom-Lie algebra). On , define an invertible linear map ψ by , where satisfies PQ = kQP, for some k ∈ ℂ, and then ψ is an Iso-automorphism of .

(5) Let be a Hom-Lie algebra with

()

If Q = ±P is satisfied, then is isotopic to ,.

(6) Let and 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).

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

Abstract

1. Introduction

2. Hom-Lie Algebras

3. Iso Algebras

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

Acknowledgment

References

References

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

Abstract

1. Introduction

2. Hom-Lie Algebras

3. Iso Algebras

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

Acknowledgment

References

References

Related

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