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**Da Motta Ferreira JC ^{*} and Bruno Marietto MG**

Center for Mathematics, Computation and Cognition, Federal University of ABC, 09210-170 Santo André, SP, Brazil

- Corresponding Author:
- Da Motta Ferreira JC

Center for Mathematics

Computation and Cognition

Federal University of ABC

09210-170 Santo André, SP, Brazil

**Tel:**+551149967914

**E-mail:**[email protected]

**Received date:** April 23, 2015; **Accepted date:** November 11, 2015; **Published date:** November 14, 2015

**Citation:** Da Motta Ferreira JC, Bruno Marietto MG (2015) Generalizing Two Structure Theorems of Lie Algebras to the Fuzzy Lie Algebras . J Generalized Lie Theory Appl 9:234. doi:10.4172/1736-4337.1000234

**Copyright:** © 2015 Da Motta Ferreira JC, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this paper we generalize two structure theorems of the class of Lie algebras to the class of fuzzy Lie algebras, namely the structure theorem of semisimple Lie algebras and the Levi’s decomposition theorem. Some open questions are also given.

**Semisimple fuzzy lie algebras**; **Levi’s fuzzy decomposition**; Fuzzy lie algebras

Lie algebras were proposed by Sophus Lie [1] and there are many applications of them in several branches of physics [2]. The notion of fuzzy sets was introduced by Zadeh [3] and many mathematicians have been involved in extending the concepts and results of abstract Lie algebra to fuzzy theory. This paper is the continuation of the results obtained in [4], where we presented conditions to generalize the concepts of solvable and nilpotent radicals of Lie algebras (called of solvable and nilpotent **fuzzy radicals**, respectively) to a class of **fuzzy Lie algebras**. In this article we use the solvable fuzzy radical to generalize the structure theorem of semisimple Lie algebras and the Levi’s decomposition theorem to a class of the fuzzy Lie algebras. The results presented in this paper are still strongly connected with results proved in [5-10].

In this section we present the basic concepts of fuzzy sets, fuzzy Lie algebras, fuzzy ideals among others which will be used throughout this paper. More details referring to these concepts and its properties can be found in [4].

A mapping of a non-empty set X into the closed unit interval [0,1] is called a * fuzzy set* of X and the set {

A set S ⊂ [0,1] is said to be an *upper well ordered set* if for all non-empty subsets C ⊂ S, then sup*C∈C.* One defines the set.

F(X, S) = {*v* | *v* is an fuzzy set of X such that *v*(X) ⊆ S}.

Let L be a Lie algebra over a field F. A fuzzy set *μ* of L is called a *fuzzy Lie algebra *of L if satisfies the following conditions: (i)* μ (ax + by)* ≥ *μ* (*x*) ∧ *μ* (*y*), (ii) *μ* (*xy*) ≥ *μ* (*x*) ∧ *μ* (*y*) and (iii)*μ*(0) = 1, for all *a, b*∈F and *x, y*∈L. A fuzzy set v of L is called a *fuzzy subalgebra *of *μ* if v is a fuzzy Lie algebra of L satisfying v(x) ≤ *μ* (x) for all x∈L. One has that *μ* is a fuzzy Lie algebra of L if, and only if, the *t*-level sets [*μ*]_{t} are **subalgebras **of L, for all *t*∈]0,1]. Also, *v* is a fuzzy subalgebra of *μ* if, and only if, the t-level sets [v]_{t} are subalgebras of [*μ*]_{t}, for all *t*∈]0,1]. Moreover, if *μ* is a fuzzy algebra of L then *μ*^{*} is a subalgebra of L.

A fuzzy set v of L is called a * fuzzy ideal* of L if satisfies the following conditions:

We define the *null fuzzy algebra* of *μ* as the fuzzy set of L

and as a consequence of this definition we assume that our upper ordered set *S* has the real numbers 0 and 1.

A fuzzy Lie algebra μ of L is called abelian if *μ ^{2}* = 0 and

If *v _{1},…,v_{n}* are fuzzy sets of L, one defines:

To understand the main result of [4] which will be used at the end of this section and in the remainder of this article, we present the following definition.

For any fuzzy subalgebra *v* of a fuzzy algebra *μ* one defines inductively the *derived series* of *v* as the descending chain of fuzzy subalgebras of by setting *v ^{(1)} = v* and for every

**Theorem 2.1. ***[4, Theorem 24] Let *L* be a finite dimensional Lie algebra over a field *F* and *S* an upper well ordered set. Then every solvable (resp., nilpotent) fuzzy ideal v of μ in *F(L,S)* is contained in a unique maximal solvable (resp., nilpotent) fuzzy ideal of μ in *F(L,S)*, called solvable (resp., nilpotent) fuzzy radical of μ in *F(L,S) *and denoted by R (μ, S) (resp., N(μ, S)).*

Let μ be a fuzzy Lie ideal of L. One says that *μ* is a *simple fuzzy ideal* if: *(i) μ* is a non-abelian fuzzy ideal and *(ii) *for all fuzzy ideals *v* of *μ*, one has either [*v*]_{t} = [*μ*]_{t} or [*v*]_{t} = {0}, for all *t*∈]0,1].

To conclude this section, we extend the notion of a semisimple fuzzy ideal [4, Definition 26] for a semisimple fuzzy algebra.

Let L be a finite dimensional Lie algebra over a field F, S an upper well ordered set and μ a fuzzy algebra of L in F(L,S). One says that *μ* is a *semisimple fuzzy algebra* in F(L,S) if: *(i)* *μ* is a non-abelian fuzzy algebra and (ii) its solvable fuzzy radical in F(L,S) is o, that is, *R* (*μ, S*) = *o*.

In this section we generalize the theorem of decomposition of a semisimple Lie algebra as a direct sum of simple Lie ideals for the case of a semisimple fuzzy ideal, similarly to the crisp case. For this, we begin with the following definition.

**Definition 3.1. **Let L be a Lie algebra over a field F, S an upper well ordered set and *μ* a fuzzy ideal of L. One says that a fuzzy set *π* of L is a fuzzy ideal of *μ* relative to *μ*^{*} if the following conditions are satisfied:

*(i)* *π* ≤ *μ* (and hence [*π*]_{t} ⊆ [*μ*]_{t}, for all *t* ∈]0,1]);

*(ii)* [*π*]_{t} is an ideal of *μ*^{*} for all *t* ∈]0,1].

In this case, *π*^{*} is also an ideal of *μ*^{*}.

If *π* is a fuzzy ideal of *μ* relative to *μ ^{*}*, then one says that a fuzzy set

*(iii)* *σ* ≤ π (and hence [*σ*]_{t} ⊆ [*π*]_{t}, for all *t* ∈]0,1]);

*(iv)* [*σ*]_{t} is an ideal of *π*^{*} for all *t* ∈]0,1].

In this case, *σ*^{*} is also an ideal of *π*^{*}.

One says that a fuzzy ideal *π* of *μ* relative to *μ*^{*} is a simple fuzzy ideal of *μ* relative to *μ*^{*} if the following conditions are satisfied:

*(v)* *π* is a non-abelian fuzzy ideal in *μ*^{*} (that is, *π ^{2}* ≠ o in

*(vi) *for all fuzzy ideal *σ* of *π* relative to *μ*^{*}, one has either [*σ*]_{t} = [*π*]_{t} or [*σ*]_{t} = {0} for all *t* ∈]0,1].

Hereafter, we exemplify each of the fuzzy ideals defined above.

**Example 3.2.** Let L be a finite dimensional Lie algebra over a field F, M is an ideal of L and *μ* a fuzzy set of L defined by its *t*-level sets as: [*μ*]_{0} = L, [*μ*]_{t}= M, for all *and * *[μ] _{t}*= {0}, for all . It is easy to check that

**Theorem 3.3. ***Let L be a finite dimensional Lie algebra over a field F, μ a fuzzy Lie ideal of L and π a non-abelian (in μ ^{*}) fuzzy ideal of μ relative to μ^{*}. If π is a simple fuzzy ideal of μ relative to μ^{*}, then π^{*} is not a solvable ideal of μ^{*}.*

*Moreover, π is a simple fuzzy ideal of μ relative to μ ^{*} if, and only if, π^{*} is a simple ideal of μ^{*}.*

**Proof.** First, let us observe that (*π ^{*}* )

Now let us consider J an ideal of *π ^{*}*. Again, by Negoita-Ralescu representation [7, Theorem 2.10], let us consider the fuzzy set

**Theorem 3.4. ***Let L be a finite dimensional Lie algebra over a field F of the characteristic *0*, S an upper well ordered set and μ a non-abelian fuzzy ideal of L in *F(L,S)*. Then μ is semisimple in *F(L,S) *if, and only if, μ*^{*}* is a semisimple ideal of L.*

**Proof.** Firstly, let us observe that , where R(*μ*^{*}) and R(L) are the radicals of *μ*^{*} and L, respectively, by [12, Theorem 3.7]. Since R(*μ*^{*}) is a solvable ideal of L contained in* μ*^{*}, then R(*μ*^{*}) = {0}, by [4, Theorem 5.6]. Thus *μ*^{*} is a semisimple ideal of L. Reciprocally, is immediate that R*(μ, S) = o,* by [12,Theorem 3.7] again.

**Theorem 3.5.*** Let *L* be a finite dimensional Lie algebra over a field* F* of the characteristic 0, S an upper well ordered set and μ a non-abelian fuzzy ideal of *L* in *F(L,S)*. If μ is semisimple in *F(L,S)*, then there are simple fuzzy ideals v _{1},…,v_{n} of μ relative to μ^{*} in *F(L,S)

*Moreover:*

*(i) for every simple fuzzy ideal π of μ relative to ^{*} there is a unique fuzzy ideal v_{i} (1 ≤ i ≤ n) such that π^{*} = ν^{*}_{i} *

*(ii) for each fuzzy ideal π of μ relative to μ ^{*} there are fuzzy ideals such that .*

**Proof. **From the hypothesis of the Theorem, we have that *μ ^{*}* is an ideal of L which is a semisimple algebra, by Theorem 3.4. Hence,

where or *μ ^{*}*

Next, let us show that . For an arbitrary x ∈ L, let us take (i) = α. If α = 0, then x ∉ *μ ^{*}*.Hence for all sum , there is at least one index i such that which implies

Now, let *π* be a simple fuzzy ideal of μ relative to *μ ^{*}*. Then

In this section we generalize the theorem of Levi decomposition of a Lie algebra for the case of a class of fuzzy ideal, similarly to the crisp case.

**Definition 4.1.** Let L be a finite dimensional Lie algebra over a field F, S an upper well ordered set and *μ* a fuzzy ideal of L in F(L,S). One says that *μ* has Levi’s hereditary if there is a semisimple subalgebra *v ^{*}* of L such that:

*(i) ; *

(ii) * * for all t ∈]0,1].

In the following example we show that the conditions of the Definition 4.1 are not artificial.

**Example 4.2.** Let L be a finite dimensional Lie algebra over a field F of the characteristic 0, * * and μ a fuzzy set of L defined by its t-level sets as: [μ]_{t} = L for all * *and [*μ*]_{t} = {0} for all *. * It is easy to check that S is an upper well ordered set and μ a fuzzy ideal of L in F(L, S) such that *μ ^{*}*= L. Also, [R(

**Theorem 4.3.*** Let *L* be a finite dimensional Lie algebra over a field *F, S* an upper well ordered set and μ a fuzzy ideal of *L* in *F(L,S)*. If μ has Levi’s hereditary, then there exists a semisimple fuzzy subalgebra v of μ in *F(L,S)* such that*

**Proof. **From the hypothesis of the Theorem, there is a unique fuzzy set v of L such that [*vI _{t}* = v

**Example 4.4. **From the Theorem 4.3 we can conclude that in the Example 4.2 there exists a semisimple fuzzy subalgebra *v* of *μ* in F(L, S) such that .

The history of the class of fuzzy Lie algebras proposed in [4] and in this paper is far from over. In fact, there are many unanswered questions and we list some of them as follows:

**Question 5.1 ***Find a fuzzy version of Malcev-Harish-Chandra’s Theorem.*

**Question 5.2** *Is it possible a fuzzy representation theory for the structure theory proposed?*

**Question 5.3** *Perspectives for applications in Physics. Reviewing the history of applications of Lie algebras in Physics since its origin, is it possible to determine methods based on the fuzzy Lie algebras presented in this work, to be applied in the Particle Physics?*

The authors would like to thank the referee for the valuable comments and suggestions.

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