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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 275250, 8 pages
http://dx.doi.org/10.1155/2013/275250
Research Article

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

Department of Mathematics, Tongji University, Shanghai 200092, China

Received 21 March 2013; Accepted 19 June 2013

Academic Editor: Shi Weichen

Copyright © 2013 Xiuxian Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We construct classical Iso-Lie and Iso-Hom-Lie algebras in 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 , where 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 [13] is the generalization of the unit to a new one called isotopic unit or isounit. The resulting new mathematical structures include the old ones as special cases.

Hom-Lie algebras 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 , 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 [79].

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 , in which the binary operation satisfies an twisted version of associativity. The authors mentioned that Lie-Santilli isotopies [13] 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 is better understood now [11, 12]. Classical simple Lie algebras over can be classified into , , , types. Similarly, we can use the twisted commutator bracket through Iso-deformation to study classical Iso-(Hom-)Lie algebras over .

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 , , , -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 “” the identity matrix, “” the linear group, and “” the transpose of matrix , and “” denotes the isomorphism of algebras. Without otherwise stated, all algebras are finite dimensional and over the complex field .

2. Hom-Lie Algebras

Definition 1 (see [4]). A Hom-Lie algebra is a triple consisting of a vector space over , a linear sef-map , and a bilinear map satisfying

When , is degenerated to Lie algebra . A Hom-Lie algebra is called multiplicative if is satisfied [13]. We call Lie-type if there exists a Lie algebra such that ,. At the same time is called the Hom-Lie admissible algebra of .

A subspace of is called an ideal of if , are satisfied. We call a Hom-Lie algebra simple if it has no nontrivial ideals and .

An automorphism of a Hom-Lie algebra is an invertible linear self-map satisfying , and .

An isomorphism of Hom-Lie algebras with is an invertible linear map satisfying and .

Lemma 2. Let be a multiplicative simple Hom-Lie algebra. Then is invertible.

Proof. It is easy to check that is an ideal of . By the simplicity of , we have . That is, is invertible.

Lemma 3 (see [14]). Let be a Lie algebra over with an algebraic homomorphism . Define a bracket by , . Then is a Hom-Lie algebra.

Theorem 4. Let be a multiplicative Hom-Lie algebra with invertible. Then is Lie-type with the Hom-Lie admissible algebra , where is defined by .

Proof. Define . It is easy to check that Now we prove that is a Lie algebra. The skew-symmetricity of is obvious. , here denotes the cyclic summation over . We have the conclusion.

Lemma 5 (see [15]). Let and be Lie type Hom-Lie algebra with invertible. Then an invertible linear map is an isomorphism of Hom-Lie algebras if and only if is an isomorphism of their Hom-Lie admissible algebras satisfying .

Lemma 6 (see [15]). Let be a Lie type Hom-Lie algebra with invertible. Then is an automorphism of if and only if is an automorphism of the Hom-Lie admissible algebra satisfying .

Proposition 7. Let be a finite dimensional Lie type Hom-Lie algebra with invertible. If its Hom-Lie admissible algebra is simple, then is simple.

Proof. By Theorem 4, the Hom-Lie admissible algebra can be written as , where is defined by . Suppose , then which is a contradiction with the simplicity of . This reduces to .
Let be a nontrivial ideal of . By definition there are ; . Therefore, . That is, is a nontrivial ideal of the Hom-Lie admissible algebra, which is impossible. We have the conclusion.

3. Iso Algebras

Let be an associative algebra with the conventional associative product over . An Iso-associative algebra is the same vector space over with a product satisfying , where the product is defined by one of the following cases, the parameters , appearing in cases II–V are fixed not necessary belonging to (it deserves to mention that in [3], is an Iso-associative algebra over an Iso-field , specially, in this paper we study Iso-algebra over ):I:  is  fixed;II:  is  invertible;III:  ;IV:  ;V:  .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 is a vector space over with a bracket defined by , where is an Iso-associative algebra. At the same time is called the Lie admissible algebra of .

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

We call an Iso-Hom-Lie algebra if its Hom-Lie admissible algebra is an Iso-Lie algebra.

Corollary 9. Let be an Iso-Lie algebra with an Iso-automorphism . Define a bracket by . Then is an Iso-Hom-Lie algebra.

We call 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 ,  ,  , -Iso-(Hom-)Lie algebras over . Give their Iso-automorphisms and isotopies. Prove that they are simple.

Let be a finite vector space, is the linear space with the conventional associative product . is an invertible element, . is an II type Iso-Lie algebra over , with the bracket defined by . It is easy to know that the conventional linear Lie algebra over is a special II type Iso-Lie algebra when is degenerated to the trivial identity.

Theorem 10. Let be a vector space satisfying (1) Define a bracket by . Then is an II type Iso-Lie algebra (one calls it an -Iso-Lie algebra).  (2) Define an invertible linear map by where satisfying , and then is an Iso-automorphism of .(3) There is an isotopy between the conventional -type Lie algebra and . Moreover, is simple.(4) Let and be -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 -Iso-Hom-Lie algebra) with an Iso-automorphism defined by , where satisfying and , for some . (6) Let be a Hom-Lie algebra with If , for some is satisfied, then is isotopic to . (7) Let and be -Iso-Hom-Lie algebras. They are isotopic if and only if is conjugate with (for some ).

Proof. (1) , there are ; , then ; therefore, . The Jacobi identity can be checked directly. By Definition 8, ,   is an II-type Iso-Lie algebra.
(2) Because , so and are satisfied. Consider , Therefore .  , So is an Iso-automorphism of .
(3) Define an invertible linear map by It is obvious that is invertible and ; therefore, . We have the first conclusion.
Suppose is a nontrivial ideal of . Let . Then Therefore, is a nontrivial ideal of , which is a contradiction with the simplicity of . We have that is simple.
(4) Define an invertible linear map 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 is an Iso-automorphism of ,  , by Lemma 6, is an Iso-automorphism of the Hom-Lie admissible algebra and satisfying
By (2) of the theorem again we have , where satisfies . Consider , and (17) is equivalent to
By (3) of the proof, there exists such that . Equation (18) is equivalent to
By the arbitrariness of , we have for all , (19) is established. According to Schur's lemma we have , for some .
(6) If is an isotopy from to ,  ; according to Lemma 5, is an isotopy of their Hom-Lie admissible algebras and satisfying
By (3) of the theorem, can be defined as . Then , (20) is equivalent to By Schur's lemma, we have , for some .
(7) According to (6) of the theorem, Then Suppose is an isomorphism of and , and then is an automorphism of satisfying By Lie theory can be defined by , , . Then (24) is equivalent to By Schur's lemma we have That is, and are conjugate (for some ).

Theorem 11. Let be a -dimensional vector space with a nondegenerate skew symmetric bilinear form . A subspace of is set
(1) On , define a bracket by . Then is a II type Iso-Lie algebra (we call it an -Iso-Lie algebra). Define an invertible linear map by , where satisfying ;  . Then is an Iso-automorphism of . Furthermore, is a group (we call it the Iso-symplectic group).
(2) Let be a conventional symplectic Lie algebra. Define an invertible linear map by . Then is an isotopy from to . Furthermore, is simple.
(3) Let and be -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 -Iso-Hom-Lie algebra). Define an invertible linear map of by , where   satisfies , for some ; then is an Iso-automorphism of ,  .
(5) Let be a Hom-Lie algebra with
If is satisfied, then is isotopic to ,  .
(6) Let and be -Iso-Hom-Lie algebras. They are isotopic if and only if and are conjugate (for some ).

Proof. (1) Let, So . The Jacobi identity can be checked directly. Hence is an II-type Iso-Lie algebra.
On , define an invertible linear map by ,  , where satisfying , we have So ,
Therefore, is an Iso-automorphism of . Let , Hence . It is obvious that is a unit of and for every element , there is an invertible element . We have that is a group.
(2)   Because , can be checked as in (3) of Theorem 10. Therefore, is an isotopy from to . is simple and can be proved as (3) of Theorem 10.
(3)  Define an invertible linear map by . Then can be proved as (2) of the proof. Therefore, is an isotopy from ,   to .
(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 is an Iso-automorphism of , and according to Lemma 6, is an Iso-automorphism of the Hom-Lie admissible algebra satisfying By (1) of the theorem, can be defined by , where . So (37) is equivalent to By (2) of the proof, such that . Then (38) is equivalent to By the arbitrariness of , we have for all , (39) is established. According to Schur's lemma, we have , for some .
(5) Suppose is an isotopy from to ,  . According to Lemma 5, is an isotopy of their Hom-Lie admissible algebras satisfying By (2) of theorem, can be defined by . Equation(40) is equivalent to By Schur's lemma we have , for some . According to (4) of the theorem, , so . Because , then . We have the conclusion.
(6) The same reason as (7) of Theorem 10, ,   and are isotopic if and only if . Suppose is an isomorphism from to , then is an automorphism of    satisfying By Lie theory we know can be defined by , where   and (42) is equivalent to By Schur's lemma we have . That is, is conjugate with (for some ).

Theorem 12. Let be an -dimensional vector space with a nondegenerate symmetric bilinear form . Define a subspace of by
(1) On define a bracket by , and then is an II type or Iso-Lie algebra.
Define an invertible linear map by , where satisfying . Then is an Iso-automorphism of . Furthermore, is a group (we call it the Iso-orthogonal group).
(2) Let be a conventional orthogonal Lie algebra. On , define an invertible linear map by . Then is an isotopy from the conventional Lie algebra to . Furthermore is simple.
(3) Let and be (or )-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 or -Iso-Hom-Lie algebra). On , define an invertible linear map by , where satisfies , for some , and then is an Iso-automorphism of .
(5) Let be a Hom-Lie algebra with
If is satisfied, then is isotopic to ,.
(6) Let and be (or )-Iso-Hom-Lie algebras. They are isotopic if and only if and are conjugate (for some ).

Acknowledgment

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

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