Extraction Algorithm of Hom-Lie Algebras Based on Solvable and Nilpotent Groups

Hom-Lie algebras are generalizations of Lie algebras that arise naturally in the study of nonassociative algebraic structures. In this paper, the concepts of solvable and nilpotent Hom-Lie algebras studied further. In the theory of groups, investigations of the properties of the solvable and nilpotent groups are well-developed. We establish a theory of the solvable and nilpotent Hom-Lie algebras analogous to that of the solvable and nilpotent groups. We also provide examples to illustrate our results and discuss possible directions for further research.


INTRODUCTION
The study of solvable and nilpotent groups has a long and rich history that dates back to the early days of group theory.The first examples of solvable groups were discovered by Évariste Galois in the 19th century, who used them to study the roots of polynomial equations.In the early 20th century, Camille Jordan and Felix Klein introduced the modern definitions of solvable and nilpotent groups, respectively.
In the mid-20th century, the theory of solvable and nilpotent groups gained importance in the context of finite group theory, particularly in the classification of finite simple groups.The classification theorem for finite simple groups, completed in 1983, relies heavily on the theory of solvable and nilpotent groups.
In the latter half of the 20th century, the study of solvable and nilpotent groups expanded to include Key words and phrases.Hom-Lie algebras, solvable Hom-Lie algebra, nilpotent Hom-Lie algebra, multiplicative algebra.infinite groups and their applications in geometry, topology, and number theory.Notable contributions include the work of John Milnor on the homology of solvable Lie groups and the study of nilpotent Lie algebras in the context of algebraic geometry and string theory.
Today, the theory of solvable and nilpotent groups remains an active area of research, with connections to a wide range of fields in mathematics and physics.Researchers continue to explore the deep connections between these groups and other areas of mathematics, paving the way for new insights and discoveries in the years to come.
There is a close relationship between solvable and nilpotent groups and solvable and nilpotent Lie algebras.In fact, the concepts of solvable and nilpotent Lie algebras were developed specifically to study the structure of solvable and nilpotent Lie groups.
Given a Lie group, one can associate a Lie algebra to it by considering the tangent space at the identity element.This Lie algebra inherits many of the properties of the original group, including its solvability and nilpotence.
More specifically, a Lie group is solvable if and only if its Lie algebra is solvable.Similarly, a Lie group is nilpotent if and only if its Lie algebra is nilpotent.
The correspondence between Lie groups and Lie algebras also allows for the translation of many results between the two contexts.For example, the Lie-Kolchin theorem states that a solvable algebraic group over an algebraically closed field has a triangular matrix representation.This result can be translated into the language of Lie algebras to obtain a similar statement for solvable Lie algebras.
Overall, the study of solvable and nilpotent groups and Lie algebras is intimately connected, with each providing insights into the other.This relationship has led to significant advances in both areas of mathematics, as well as applications in physics and other fields.
The hom-Lie algebras which are generalizations of classical Lie algebras was constructed by by Hartwig, Larsson, and Silvestrov [5] in 2006.Since then, many mathematicians have been trying to extend known results in the setting of Lie algebras to the setting of hom-Lie algebras (see e.g.[7,9,10,11]).Homo-Lie algebras have received a lot of attention lately because of their close connection to discrete and deformed vector fields and differential calculus [5,15,16].
In the present article, we study solvable and nilpotent hom-Lie algebras, which can be viewed as an extension of solvable and nilpotent Lie algebras.

PRELIMINARIES
he following is a definition from [14] with F denoting a ground field: We consider two Hom-Lie algebras (L 1 , [ , ] 1 , α 1 ) and (L 2 , [ , ] 2 , α 2 ), and define a linear map ϕ : L 1 → L 2 .If ϕ satisfies the following two conditions, then it is called a morphism of Hom-Lie algebras: a bijective morphism of Hom-Lie algebras, it is referred to as an isomorphism of Hom-Lie algebras.In this case, we say L 1 and L 2 are isomorphic and write Furthermore, a subspace H of L is called a Hom-Lie subalgebra if α(x) ∈ H and [x, y] ∈ H for all x, y ∈ H.If [x, y] ∈ H holds true for all x ∈ H and y ∈ L, then H is called a Hom-Lie ideal.
algebra, where the Hom-bracket operation [ , ] is defined by and the linear operator is defined as Example 2.5.( [7]) Let F = C be the field of complex numbers.Consider the vector space C 2 and define the linear map We define the bilinear map [ , ] * : and the skew-symmetric bilinear map where Hom-Lie algebra.
Example 2.7.We can make (R[x], [, ], α) a Hom-Lie algebra, where R[x] is the vector space of polynomials with coefficients in R, and α : R[x] → R[x] is the linear map defined by α(p(x)) = p(0) It can be verified that [•, •] is antisymmetric and satisfies the Hom-Jacobi identity, which makes For p(x), q(x), h(x) ∈ R[x], then one can easily see that Thus, for each p(x), q(x), h(x) ∈ R[x] we have ) is not a Lie algebra, since The following theorem, as presented in the publication by Casas [3], lacks a formal proof.Then, (ii) [H, K] is a Hom-Lie ideal of H and K, respectively.
(iii) [H, K] is a Hom-Lie ideal of L when α is onto. Proof.
Similarly, [H, K] is also a Hom-Lie ideal of K.
(iii) As per (i), [H, K] is a Hom-Lie subalgebra of L. Therefore, it suffices to prove that

✷
The subsequent example demonstrates that Theorem 2.1 (iii) is invalid if α is not a surjective map.
is a morphism of Hom-Lie algebras.
α) be a multiplicative Hom-Lie algebra.We define ,{L (i) }, i ≥ 0, the derived series of L by Note that ] is a Hom-Lie ideal of L (i−1) (by induction and Theorem 2.1(ii)).
Thus the derived series is a descending series.

Definition 3.2. ([8]
) A multiplicative Hom-Lie algebra (L, [ , ], α) is said to be solvable if there Clearly a multiplicative Hom-Lie algebra is solvable of class Hom-Lie algebras is the same as in the case of Lie algebras ( [12]) are the solvable Hom-Lie algebras of class at most 2.
Note that L i+1 = [L, L i ] is a Hom-Lie ideal of L i (by Theorem 2.1(ii) and induction).
Thus the lower central series is a descending series.
Thus L is a nilpotent Hom-Lie algebras of class n − 2.

EXTRACTION ALGORITHM OF HOM-LIE ALGEBRAS BASED ON SOLVABLE AND NILPOTENT GROUPS 13
Definition 4.3.Let (L, [ , ], α) be a multiplicative Hom-Lie algebra.Then a descending series Proof.It follows directly from the definition of Proof.Applying induction we see α) be a multiplicative Hom-Lie algebra.Then L is nilpotent of class ≤ k iff there exists a central series of length k.
Proof.If L is a nilpotent Hom-Lie algebra of class ≤ k.Then Proof.A Hom-Lie algebra L is nilpotent of class 1 iff there exists a central series of length ) be a morphism of multiplicative Hom-Lie algebras.Then Proof.
Note that (R[x], [ , ], α) is not a solvable and not a nilpotent Hom-Lie algebra because there exists a non-solvable and non-nilpotent Hom-Lie subalgebra of R[x] (Theorem 3.5(i) and Theorem 4.5(i)).

Question for Further Research
Question 5.1.What are the precise conditions for a Hom-Lie algebra to be solvable or nilpotent?Can these conditions be expressed in terms of the underlying Lie algebra and the Hom morphism?These questions are just a starting point, and there are many other avenues for research in this area.By exploring these and other questions, researchers can gain a deeper understanding of the properties and applications of solvable and nilpotent Hom-Lie algebras, and advance our knowledge of this important area of algebraic research.

Conclusion
In conclusion, this paper presents an extraction algorithm for Hom-Lie algebras that is based on solvable and nilpotent groups.The algorithm involves several steps.The algorithm is illustrated with examples, which demonstrate its effectiveness in extracting Hom-Lie algebra structures.
Overall, the extraction algorithm presented in this paper provides a useful tool for studying Hom-Lie algebras, which have important applications in various areas of mathematics and physics.The

Example 2 . 1 .Example 2 . 2 .
([14]) Every Lie algebra can be considered as a Hom-Lie algebra by taking α as the identity map, i.e., α = id L .Consider a vector space L over F , equipped with an arbitrary skew-symmetric bilinear map [, ] : L × L → L, and let α : L → L denote the zero map.It follows straightforwardly that (L, [, ], α) forms a Hom-Lie algebra with multiplication.

Definition 4 . 2 .
([8]) Let (L, [ , ], α) be a multiplicative Hom-Lie algebra.We say that L is nilpotent if there exists n ∈ N such that L n = 0.It is nilpotent of class k if L k = {0} and

Example 4 . 1 .
Consider the multiplicative Hom-Lie algebra (L, [ , ], α) in Example 3.1 where α is the zero map and [ , ] is the skew-symmetric bilinear map such that [e i , e j ] = 0 if i = j or i = 1 and

Question 5 . 2 .Question 5 . 3 .Question 5 . 5 .Question 5 . 6 .Question 5 . 8 .
What are some examples of solvable Hom-Lie algebras, and what properties do they have?Are there any interesting relationships between these examples and other areas of mathematics, such as Lie theory or algebraic geometry?What are some examples of nilpotent Hom-Lie algebras, and how do they compare to nilpotent Lie algebras?Can the classification of nilpotent Lie algebras be extended to the Hom-Lie algebra setting?Question 5.4.How do solvable and nilpotent Hom-Lie algebras arise in physics, particularly in the context of supersymmetry and other quantum field theories?What are the implications of these structures for our understanding of fundamental physics?What is the relationship between Solvable and Nilpotent Hom-Lie algebras and other algebraic structures, such as associative algebras or Lie superalgebras?Can techniques from these other areas be used to study solvable and nilpotent Hom-Lie algebras more effectively?How can the representation theory of Hom-Lie algebras be studied, particularly in the case of solvable and nilpotent algebras?What are some interesting examples of Hom-Lie algebra representations, and what do they tell us about the structure of these algebras?Question 5.7.Study of Hom-Lie superalgebras: Hom-Lie superalgebras are a natural generalization of Hom-Lie algebras that incorporate a Z 2 -grading.Investigating solvable and nilpotent Hom-Lie superalgebras can lead to interesting results in the study of supersymmetry and related topics in physics.Generalization of results to other categories: Hom-Lie algebras are defined in the category of vector spaces, but similar structures can be defined in other categories, such as modules or abelian groups.Investigating solvable and nilpotent Hom-Lie algebras in these categories can provide insight into the interplay between different areas of algebra.Question 5.9.Cohomology of Hom-Lie algebras: Cohomology is a powerful tool for understanding the structure of Lie algebras, and similar techniques can be applied to Hom-Lie algebras.Investigating the cohomology of solvable and nilpotent Hom-Lie algebras can provide insights into their structure and classification.Question 5.10.Quantum Hom-Lie algebras: Quantum Hom-Lie algebras are a generalization of Hom-Lie algebras that arise in the context of quantum groups and deformation theory.Investigating solvable and nilpotent quantum Hom-Lie algebras can lead to interesting results in these areas.Question 5.11.Applications to cryptography and coding theory: Hom-Lie algebras have recently been applied to cryptography and coding theory.Investigating solvable and nilpotent Hom-Lie algebras in this context can lead to new methods for error-correction and secure communication.
called a solvable series if for each i, we have L i+1 is a Hom-Lie ideal of L i and L i /L i+1 is an abelian Hom-lie algebra.Note that, L 1 × L 2 is a multiplicative Hom-Lie algebra because L 1 and L 2 are multiplicative Hom-Lie algebras.Since L 1 Lemma 3.1.Let H be a Hom-Lie subalgebra of the Hom-Lie algebra (L, [ , ], α).Then H is a Hom-Lie ideal of L and L/H is abelian Hom-Lie algebra if and only if [L, L] ⊆ H Proof.If L/H is an abelian Hom-Lie algebra, then for any [x, y] ∈ [L, L] we find H = [x + H, y + H] = [x, y] + H. Therefore [x, y] ∈ H. Conversely, if [L, L] ⊆ H then [x, y] ∈ H for all x ∈ H (⊆ L) and y ∈ L, which implies H is a Hom-Lie ideal of L. Also, for any x, y ∈ L we have [x + H, y + H] = [x, y] + H = H (because [x, y] ∈ [L, L] ⊆ H). ✷ Proof.