Extraction Algorithm of HOM–LIE Algebras Based on Solvable and Nilpotent Groups

Abstract

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 are 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.

Dedicated to Al Farouk School & Kinder garten-Irbid-Jordan

1. 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 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 nil potency.

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 were constructed by Hartwig et al. [1] in 2006. 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., [2–5]). Hom–Lie algebras have received a lot of attention lately because of their close connection to discrete and deformed vector fields and differential calculus [1].

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 [6–9].

2. Preliminaries

The following is a definition from [1] with denoting a ground field.

Definition 1. (see [1]). A Hom–Lie algebra over is a triple consisting of a vector space over , a linear map , and a bilinear map (called a Hom–Lie bracket), which satisfies the following two conditions:(i)Skew-symmetry property: for all (ii)Hom–Jacobi identity: , for all If holds for all , then the Hom–Lie algebra is referred to as multiplicative.
We consider two Hom–Lie algebras and and define a linear map . If satisfies the following two conditions, then it is called a morphism of Hom–Lie algebras:(i) for all (ii)If is a bijective morphism of Hom–Lie algebras, it is referred to as an isomorphism of Hom–Lie algebras. In this case, we say and are isomorphic and write .
Furthermore, a subspace of is called a Hom–Lie subalgebra if and for all . If holds for all and , then is called a Hom–Lie ideal.

Example 1. (see [1]). Every Lie algebra can be considered as a Hom–Lie algebra by taking as the identity map, i.e., .

Example 2. Consider a vector space over , equipped with an arbitrary skew-symmetric bilinear map , and let denote the zero map. It follows straightforwardly that forms a Hom–Lie algebra with multiplication.

Example 3. (see [1]). Let be a vector space and be any linear operator. Then, is a Hom–Lie algebra, where for all . Such Hom–Lie algebras are referred to as abelian (commutative) Hom–Lie algebras.

Example 4. (see [6]). Suppose are Hom–Lie algebras. Then, the direct sum is also a Hom–Lie algebra, where the Hom-bracket operation is defined by the following expression:and the linear operator is defined as follows:

Example 5. (see [2]). Let be the field of complex numbers. Consider the vector space and define the linear mapWe define the bilinear map , whereThen, is a multiplicative Hom–Lie algebra.

Example 6. (see [2]). Consider the setwith the linear mapand the skew-symmetric bilinear mapwhere . Then, is a multiplicative Hom–Lie algebra.

Example 7. We can make a Hom–Lie algebra, where is the vector space of polynomials with coefficients in , and is the linear map defined by for any . We define for any by the following expression:It can be verified that is antisymmetric and satisfies the Hom–Jacobi identity, which makes a Hom–Lie algebra. Indeed, if , thenFor , then one can easily see thatThus, for each , we have the following expression:Also,It is clear that is not a Lie algebra, since

Example 8. (see [7]). Let be a Hom–Lie algebra and let be a Hom–Lie ideal. Then, the quotient space is a Hom–Lie algebra, where

Consider and as Hom–Lie ideals in a Hom–Lie algebra . We define the sum of and as the set , where . Moreover, we define the multiplication of and as the span of the set of all possible commutators between and , denoted as . Thus,

The following theorem, as presented in the publication by Casas et al. [7], lacks a formal proof.

Theorem 2 (see [7]). Let and be Hom–Lie ideals of a multiplicative Hom–Lie algebra . Then,(i) is a Hom–Lie subalgebra of (ii) is a Hom–Lie ideal of and , respectively(iii) is a Hom–Lie ideal of when is onto

Proof. (i)Let , where and . Then, . To demonstrate closure of multiplication under , we consider and in with and . Since and , it follows that .(ii)It should be noted that , as stated in . This implies that is a Hom–Lie subalgebra of both and . Furthermore, if and , then , and consequently . Thus, is a Hom–Lie ideal of . Similarly, is also a Hom–Lie ideal of .(iii)As per , is a Hom–Lie subalgebra of . Therefore, it suffices to prove that whenever and . Let , , and . Since for some , it follows that and . Hence,The subsequent example demonstrates that Theorem 2 (iii) is invalid if is not a surjective map.

Example 9. Consider the multiplicative Hom–Lie algebra , where is a vector space over with basis . The map is the zero map, and [,] is a skew-symmetric bilinear map defined as follows:and for all . Let and . It can be observed that and are Hom–Lie ideals of . However, is not a Hom–Lie ideal of , as . This example illustrates that Theorem 2 (iii) does not hold when is not onto.

Example 10. (see [2]). Let be a Hom–Lie algebra and H be a Hom–Lie ideal. Then, is a Hom–Lie algebra and the linear mapis a morphism of Hom–Lie algebras.

3. Solvable Hom–Lie Algebra

Let be a Hom–Lie algebra. The sequence of Hom–Lie subalgebras such thatis called a descending series.

Definition 3. (see [8]). Let be a multiplicative Hom–Lie algebra. We define, , the derived series of by the following expression:

Note that is a Hom–Lie ideal of (by induction and Theorem 2 (ii)).

Thus, the derived series is a descending series.

Definition 4. (see [8]). A multiplicative Hom–Lie algebra is said to be solvable if there exists such that . We say is solvable of class if and .

Clearly a multiplicative Hom–Lie algebra is solvable of class iff . Metabelian Hom–Lie algebras are the same as in the case of Lie algebras [9] which are the solvable Hom–Lie algebras of class at most 2.

Example 11. Let be the space spanned by a basis over . Consider the multiplicative Hom–Lie algebra where is the zero map and is the skew-symmetric bilinear map such that if or and if . Note that . Thus, is not a Lie algebra. Now,    If is even, then and . Thus, is a solvable Hom–Lie algebra of class . If is odd, then and . Thus, is a solvable Hom–Lie algebra of class .

Definition 5. Let be a multiplicative Hom–Lie algebra. Then, the descending series is called a solvable series if for each i, we have which is a Hom–Lie ideal of and is an abelian Hom–Lie algebra.

Lemma 6. Let be a Hom–Lie subalgebra of the Hom–Lie algebra . Then, is a Hom–Lie ideal of and is a abelian Hom–Lie algebra if and only if .

Proof. If is an abelian Hom–Lie algebra, then for any , we find . Therefore, . Conversely, if , then for all and , which implies is a Hom–Lie ideal of . Also, for any , we have (because ).

Corollary 7. Let be a Hom–Lie algebra. Then, the descending series is solvable if and only if for each .

Proof. This follows directly from Definition 5 and Lemma 6.

Theorem 8. Let be a multiplicative Hom–Lie algebra. Then, is a solvable Hom–Lie algebra of class iff is a solvable series.

Proof. This follows directly from the definition of the derived series and the corollary above.

Theorem 9. Let be a multiplicative Hom–Lie algebra. If is a solvable series, then for each i, .

Proof. We use induction. For , we have . For and because of the induction assumption, we have . Now, according to Corollary 7, we have . Therefore, .

Theorem 10. Let be a multiplicative Hom–Lie algebra. Then, is solvable of class if and only if there exists a solvable series of length .

Proof. If is a solvable Hom–Lie algebra of class , then, using Theorem 8, the series is solvable. Conversely, suppose thatis a solvable series. Then, using , we find .

Corollary 11. Solvable Hom–Lie algebras of class 1 are the abelian Hom–Lie algebras.

Proof. is solvable of class 1 iff there exists a solvable series of length 1 iff is abelian Hom–Lie algebra iff is abelian Hom–Lie algebra.

Theorem 12. Let be a morphism of multiplicative Hom–Lie algebras. Then,(i)(ii)If is solvable of class , then is solvable of class (iii)If is an isomorphism of Hom–Lie algebras, then is solvable of class if and only if is solvable of class

Proof. (i)By applying induction, we find . Also, if , then .(ii)Since is solvable of class , it follows . So, . Thus, is solvable of class .(iii)Using we may assume that and are solvable of classes and , respectively. Again by (ii) we have . Also, since is an isomorphism of Hom–Lie algebras, then is solvable of class ; that is . Thus, .

Example 12. The Hom–Lie algebras (Example 6) and (Example 5) are isomorphic Hom–Lie algebras. Sinceis an isomorphism Hom–Lie algebra ([2]). Now, , and . Thus is a solvable Hom–Lie algebra of class 2. Also, is a solvable Hom–Lie algebra of class 2.

Lemma 13. Let be a multiplicative Hom–Lie algebra and be a Hom–Lie subalgebra of . Then, , for each .

Proof. Clearly, . Using induction, if , then

Theorem 14. Let be a solvable Hom–Lie algebra of class . Then,(i)Any Hom–Lie subalgebra is solvable of class (ii)Any quotient Hom–Lie algebra of is solvable of class

Proof. (i)Let be a Hom–Lie subalgebra. Then, is multiplicative (because is multiplicative). Also, according to the lemma above, we have . Thus, .(ii)Let be a Hom–Lie ideal of . Then, so is (because is multiplicative). Consider the natural map in Example 10. According to Theorem 12 (i), , which implies

Theorem 15. Let be a multiplicative Hom–Lie algebra. If H is a solvable Hom–Lie ideal of class and is solvable of class , then is solvable of class .

Proof. According to Theorem 10, we have the following two solvable series:Consider the natural map , and let . Hence, is a Hom–Lie subalgebra of and . Therefore,is a descending series. Now, it suffices to prove that . If , then and so (Corollary 7). Therefore, for each . This shows that . Therefore,is a solvable series of length . By Theorem 10, we have which is solvable of class .
In [4], we proved that if and are Hom–Lie algebras and is a Hom–Lie ideal of , ., then is a Hom–Lie ideal of and

Theorem 16. Let and be solvable Hom–Lie algebras of class and , respectively. Then, is a solvable Hom–Lie algebra of class .

Proof. Note that, is a multiplicative Hom–Lie algebra because and are multiplicative Hom–Lie algebras. Since , so is a solvable Hom–Lie ideal (of class ) of . Also, , so is a solvable Hom–Lie algebra of class . According to Theorem 15, is a solvable Hom–Lie algebra of class .

4. Nilpotent Hom–Lie Algebra

Definition 17. (see [8]). Let be a multiplicative Hom–Lie algebra. We define, , the lower central series of by , , and .

Note that is a Hom–Lie ideal of (by Theorem 2 (ii) and induction).

Thus, the lower central series is a descending series.

Definition 18 (see [8]). Let be a multiplicative Hom–Lie algebra. We say that is nilpotent, if there exists such that . It is nilpotent of class if and .

It is clear now that is nilpotent of class iff .

Example 13. Consider the multiplicative Hom–Lie algebra in Example 11 where is the zero map and is the skew-symmetric bilinear map such that if or and if . Now,       Thus, is a nilpotent Hom–Lie algebra of class .

Definition 19. Let be a multiplicative Hom–Lie algebra. Then, a descending series is said to be central, if for each , . It has a length if but .

Theorem 20. Let be a multiplicative Hom–Lie algebra. Then, is a nilpotent Hom–Lie algebra of class iff is a central series.

Proof. It follows directly from the definition of .

Theorem 21. Let be a multiplicative Hom–Lie algebra. If is a central series, then for each , .

Proof. Applying induction we see . Also, if then .

Theorem 22. Let be a multiplicative Hom–Lie algebra. Then, is nilpotent of class iff there exists a central series of length .

Proof. If is a nilpotent Hom–Lie algebra of class . Then,is a central series. The converse is true, since so .

Corollary 23. Nilpotent Hom–Lie algebras of class 1 are the abelian Hom–Lie algebras.

Proof. A Hom–Lie algebra is nilpotent of class 1 iff there exists a central series of length 1, iff iff iff is an abelian Hom–Lie algebra.

Theorem 24. Let be a morphism of multiplicative Hom–Lie algebras. Then,(i)(ii)If is nilpotent of class , then is nilpotent of class (iii)If is an isomorphism of Hom–Lie algebras, then is nilpotent of class if and only if is nilpotent of class

Proof. (i)We note that . Also if , then .(ii)Since is nilpotent of class , then . So, . Thus, is nilpotent of class .(iii)Let be nilpotent of class . By (ii) is nilpotent of class . Let be nilpotent of class . Since is an isomorphism of Hom–Lie algebras, it follows is solvable of class . Thus, .

Example 14. Consider Example 12. Since , and , . Thus, is not a nilpotent Hom–Lie algebra and so is not a nilpotent Hom–Lie algebra.

Lemma 25. Let be a multiplicative Hom–Lie algebra and be a Hom–Lie subalgebra of . Then, , for each .

Proof. , and by using induction, if , then .

Theorem 26. Let be a nilpotent Hom–Lie algebra of class k.(i)Any Hom–Lie subalgebra is nilpotent of class (ii)Any quotient Hom–Lie algebra of is nilpotent of class

Proof. (i)A Hom–Lie subalgebra of a multiplicative Hom–Lie algebra is multiplicative. By the lemma above, we have . Thus, .(ii)Let be a Hom–Lie ideal of . The Hom–Lie algebra is multiplicative. Consider the natural morphism . According to Theorem 24 (i), , which implies .

Remark 27. Let be a multiplicative Hom–Lie algebra. If H is a nilpotent Hom–Lie ideal and is a nilpotent Hom–Lie algebra, then need not be a nilpotent Hom–Lie algebra.

Example 15. Let be the space spanned by a basis over . Consider the multiplicative Hom–Lie algebra where is the zero map and is the skew-symmetric bilinear map such that and . Let . Then, is a nilpotent Hom–Lie ideal, because . Also, is a nilpotent Hom–Lie algebra, because . But is not a nilpotent Hom–Lie algebra, since , and for all .

Theorem 28. Let and be nilpotent Hom–Lie algebras of class and , respectively. Then, is a nilpotent Hom–Lie algebra of class .

Proof. We use induction to show that . For , we have . For and because of the induction assumption, we have .
We may assume that . Since and are nilpotent Hom–Lie algebras of class and , respectively, then and and .
Now, and . Thus, is a nilpotent Hom–Lie algebra of class .

Theorem 29. Every central series is a solvable series.

Proof. Let be a Hom–Lie algebra and be a central series. Then, for each , . Since , so is a solvable series (Theorem 8).

Corollary 30. Every nilpotent Hom–Lie algebra is a solvable Hom–Lie algebra.

Proof. If is a nilpotent Hom–Lie algebra, then there exists a central series (Theorem 22). From the theorem, is a solvable series. Thus, is a solvable Hom–Lie algebra (Theorem 10).
The converse is not true, as in the following example.

Example 16. Consider the Hom–Lie algebra in Example 7. It is easy to show that is a Hom–Lie subalgebra of . For any , , where , and so and . Thus, is a solvable Hom–Lie algebra of class 2. But is not a nilpotent Hom–Lie algebra, since , , and for all .

Example 17. Consider the Hom–Lie subalgebra of in Example 7. For any , , where , and so , , and for all . Thus, is not a solvable Hom–Lie algebra. Also, is not a nilpotent Hom–Lie algebra by Corollary 30.
Note that, is not a solvable and not a nilpotent Hom–Lie algebra because there exists a nonsolvable and non-nilpotent Hom–Lie subalgebra of (Theorems 14 (i) and 26 (i)).

5. Question for Further Research

Question 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?

Question 2. 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?

Question 3. 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 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?

Question 5. What is the relationship between solvable and nilpotent Hom–Lie algebras and other algebraic structures, such as associative algebras or Lie super algebras? Can techniques from these other areas be used to study solvable and nilpotent Hom–Lie algebras more effectively?

Question 6. 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 7. Study of Hom–Lie super algebras: Hom–Lie super algebras are a natural generalization of Hom–Lie algebras that incorporate a -grading. Investigating solvable and nilpotent Hom–Lie super algebras can lead to interesting results in the study of supersymmetry and related topics in physics.

Question 8. 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 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 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 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.

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.

6. 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 algorithm is particularly well-suited for Hom–Lie algebras that are related to solvable and nilpotent groups, which are important classes of groups that arise in many different contexts.

Further research could be done to investigate the effectiveness of the extraction algorithm for Hom–Lie algebras that are not related to solvable or nilpotent groups and to explore its potential applications in other areas of mathematics and physics. Nevertheless, the algorithm presented in this paper is a valuable contribution to the study of Hom–Lie algebras and provides a useful framework for further investigation of these important algebraic structures.

Data Availability

No underlying data were collected or produced in this study.

Disclosure

The initial preprint version of this research are made available on ArXiv [14].

Conflicts of Interest

The authors declare that there are no conflicts of interest.