Locally Finite Associative Algebras and Their Lie Subalgebras

An infinite dimensional associative algebra A over a field F is called locally finite associative algebra if every finite set of elements is contained in a finite dimensional subalgebra of A . Given any associative algebra A over field F of any characteristic. Consider a new multiplication on A called the Lie multiplication which defined by [a, b] = ab − ba for all a, b ∈ A, where ab is the associative multiplication in A . Then L = A (−) together with the Lie multiplication form a Lie subalgebra of A . It is natural to expect that the structures of L and A are connected closely. In this paper, we study and discuss the structure of infinite dimensional locally finite Lie and associative algebras. The relation between them, their ideals and their inner ideals is considered. A brief discussion of the simple associative algebras and simple Lie algebras is also be provided.


Introduction
Throughout this paper, unless otherwise stated, is an algebraically closed field of characteristic positive characteristic , is an infinite dimensional locally finite associative algebra over and is an infinite dimensional locally finite Lie algebra over .
In 2004, Bahturin, Baranov and Zalesski [1] studied simple locally finite Lie subalgebra of the locally finite associative ones. A locally finite (Lie or Associative) algebra is an algebra in which for every finite set of elements of is contained in a finite dimensional subalgebra of . The Lie structure of associative rings or algebras were investigated by the American Mathematician Herstein in 1954 (see [20] and [21]) after defining a new multiplication called the Lie Multiplication by [ , ]: = − for all , ∈ , (1.1) where is the usual associative multiplication in the simple associative ring over its centre ( ). Then ( ) together with the multiplication in (1.1) form a Lie algebra over ( ). We denote by ( ) = [ , ] to be the Lie subalgebra of ( ) together with the multiplication defined in (1.1).
Moreover, if an involution * is defined on , then for any subalgebra of skew( ): ={ ∈ : * = − } (1.2) form a Lie algebra with the Lie multiplication that defined as (1.1). Recall that an involution * : → is an anti-automorphism, defined by * ( ) = * , satisfy the following conditions * ( + ) = * + * , * ( ) = * * and * ( * (( )) = for all , ∈ . Involutions of the first kind only is considered in this paper, that is, involutions with the following property: * ( ) = * .  [17] studied the Lie subalgebras of prime rings with involutions in 1972. A revision to Herstein's Lie theory was giving by Martindale [22] 1986. All of these studies focused on the structure of the Lie ideals and Lie subalgebras that obtained from simple associative rings or algebras. Recall that a subspace of is called a subalgebra of if ( ) ⊆ and an ideal if [ , ] ⊆ . Although simple Lie algebras have no ideals except themselves and the trivial ones, it has been proved in [12] that all simple Lie algebras of classical type have non-zero inner ideals.
In 1976, the American mathematician Georgia Benkart introduced the notion inner ideals of Lie algebras. An inner ideal is a vector subspace of which satisfies the property [ , [ , ]] ⊆ . By the definition of the Lie ideals, one can see that every ideal is an inner ideal. However, Inner ideals are more difficult to be studied as some of them are even not Lie subalgebras. Benkart showed that the structure of the Lie inner ideals are similar to the structure of the -nilpotent elements of Lie algebras [13]. Therefore, inner ideals are important in classifying Lie algebras because by using certain restriction on the -nilpotent elements one can distinguish the simple Lie algebras of classical type and of the non-classical ones in the case when > 2. In several papers (See for example [14], [15] [18] and [19]) Fernández López et al generalized Benkart's theory over inner ideals.
In this paper, we discuss the structure of the infinite dimensional simple locally finite algebras. We start Section 2 with some preliminaries. Section 3 stats some facts about the plain, diagonal and nondiagonal modules of finite dimensional Lie algebra and Section 4 consists of the infinite dimensional case where the some types of local systems of locally finite algebras (associative or Lie) are considered. Section 5 is the completion of Section 3 where the infinite dimensional cases of plain diagonal and non-diagonal Lie algebras are highlighted. In Section 6 we investigate the structure of (involution) simple and associative algebras. The main results of this paper are found in Sections 7 and 8, where the simple locally finite Lie algebras of simple and involution simple associative algebras are considered.

Preliminaries
A perfect Lie algebra is a Lie algebra with the property ( ) = and a perfect associative algebra is an associative algebra such that = [4].

Definition 2.1. [1]
A locally finite (associative, Lie,…etc) algebra is an algebra (associative, Lie,…etc) over a field in which for every finite set of elements in we can find a finite dimensional subalgebra of that contained it.
Recall that a set is said to be a directed partially ordered set if there is an ordering relation ≤ defined on such that for each , ∈ , there is ∈ such that , ≤ [2].

Remark 2.2.
Suppose that for each , ∈ with ⊆ we set ≤ . Then for each , ∈ , there is ∈ such that , ≤ , so is a directed partially ordered set. Thus, lim ⟶ is the direct limits of an infinite chain of algebras ( ⊂ ⊂ ⋯ ⊂ ⊂ …). Therefore, is the inductive limit = lim ⟶ of the algebras .
We denote by ℳ ( ) the vector space of all × -matrices together with the matrix multiplication defined on it. Example 2.8. [3] Consider the locally finite associative algebra ℳ ( ) in Example 2.4. We construct three locally finite Lie subalgebras of ℳ ( ). Those are the stable special linear ( ), stable Symplectic ( ) and stable Orthogonal ( ) Lie subalgebras of ℳ ( ) that defined to be the union (or the direct limit) of the natural embeddings, respectively, Definition 2.9. [2] A locally finite (associative or Lie) algebra over a field is said to be locally semi(simple) in the case when for every finite set of elements of we can find a finite dimensional (semi)simple subalgebra of which contains . is an integer number (because is algebraically closed). Note that each embedding ⊆ is written as follows: ↦ diag( , … , , 0, … , 0), ∈ ℳ ( ).

Plain, diagonal and non-diagonal modules of finite dimensional Lie algebras.
Suppose that is perfect. Then there is a Levi (maximal semisimple) subalgebra of such that = ⊕ ℛ, where ℛ is a solvable radical of (Levi-Malcev Theorem). As ℛ is an ideal of , we have /ℛ ≅ . Let be a simple -module. Since is perfect, ( ) annihilates , so = (because is simple). Let , … , be the simple ideals of such that = ⊕ … ⊕ . Then is a completely reducible -module and = ⊕ … ⊕ , where is a simple -module.

Definition 3.2.
Suppose that is perfect and finite dimensional. Let be an -module. 1. Suppose that ≅ ( ) for each 1 ≤ ≤ . Then is said to be a plain -module if each is a natural -module. 2. Suppose that ′ is a perfect Lie algebra such that ′ is finite dimensional. Let , … , be natural -modules. An embedding ⊆ is called a plain embedding if ( ⊕ … ⊕ ) ⇂ is a plainmodule. where the integers ℓ and do not depend on and + ℓdim = dim .

Definition 3.5.
Suppose that is perfect and finite dimensional. Let be an -module. 1. Suppose that ≅ ( ), ( ), ( ) for each 1 ≤ ≤ . Then is said to be a diagonalmodule in the case when each is either a natural or a dual to natural -module. Otherwise, is said to be a non-diagonal -module. 2. Suppose that ′ is a perfect Lie algebra such that ′ is finite dimensional. Suppose that , … , are natural -modules. An embedding ⊆ is called diagonal embedding if ( ⊕ … ⊕ ) ⇂ is diagonal. where + (ℓ + )dim = dim .

Proposition 3.8. [9]
Let be a simple Lie algebra of rank greater than 10. Suppose that ⊆ ⊆ , where are all perfect and finite dimensional Lie algebras. Suppose that = 0 and ⊆ is a non-diagonal embedding. If ⇂ is non-trivial for every -module , then there is a naturalmodule such that ⇂ is a non-diagonal -module.  (3.iii.) implies that the rank of every simple ideal of any Levi (maximal semisimple) subalgebra of (for every ∈ ) is greater than or equal to the rank of . is an ideal of with ⊆ ′. The simplicity of implies that = = ⋃ ∈ . Thus, { } ∈ is a local system of . Let ∈ . Since is a finite dimensional, there exists ∈ such that ⊆ , but ⊆ . Therefore, for each ∈ , there is ∈ such that ⊆ , as required. ∎

Plain, diagonal and non-diagonal locally finite Lie algebras
Let ⊆ ′ be perfect Lie algebras. If and ′ are finite dimensional and , … , are naturalmodules, then an embedding ⊆ is called a plain (resp. diagonal) embedding if ( ⊕ … ⊕ ) ⇂ is a plain (resp. diagonal) -module.
Definition 5.1. [5] Suppose that is simple. Then a plain (resp. diagonal) local system of is a perfect local system { } ∈ such that the embedding ⊆ is plain (resp. diagonal) for all ≤ . where is a trivial and one dimensional -module and * is the dual to .
Remark 5.3 [4] Suppose that { } ∈ is a conical system of . Then all simple components of (for each ∈ ) are of classical type if the rank of is greater than or equal to 9.