[Paper Review] A Classification of Real Indecomposable Solvable Lie Algebras of Small Dimension with Codimension One Nilradicals

Alan R. Parry|arXiv (Cornell University)|Nov 24, 2013

Magnetism in coordination complexes2 references16 citations

TL;DR

This paper presents a complete classification of real indecomposable solvable Lie algebras of dimension 2 through 7 with codimension one nilradicals. Using automorphisms of the nilradical and canonical forms of derivations, the author reduces structure constants to simplest possible forms, yielding explicit multiplication tables for all isomorphism classes. The key contribution is a systematic, computationally verified classification of these algebras, with results compiled in appendices for dimensions 2–7.

ABSTRACT

This thesis was concerned with classifying the real indecomposable solvable Lie algebras with codimension one nilradicals of dimensions two through seven. This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Lie algebras as well as a few helpful theorems that are necessary to understand the project. We also reviewed many concepts from linear algebra that are essential to the research. The second chapter was occupied with a description of how we went about classifying the Lie algebras. In particular, it outlined the basic premise of the classification: that we can use the automorphisms of the nilradical of the Lie algebra to find a basis with the simplest structure equations possible. In addition, it outlined a few other methods that also helped find this basis. Finally, this chapter included a discussion of the canonical forms of certain types of matrices that arose in the project. The third chapter presented a sample of the classification of the seven dimensional Lie algebras. In it, we proceeded step-by-step through the classification of the Lie algebras whose nilradical was one of four specifically chosen because they were representative of the different types that arose during the project. In the appendices, we presented our results in a list of the multiplication tables of the isomorphism classes found.

Motivation & Objective

• To classify all real indecomposable solvable Lie algebras of dimension 2 through 7 that have a nilradical of codimension one.
• To develop a systematic method for simplifying structure constants of such Lie algebras using automorphisms of the nilradical.
• To identify and enumerate all isomorphism classes of these Lie algebras by reducing derivation matrices to canonical forms.
• To provide explicit multiplication tables for all resulting isomorphism classes in a comprehensive, computationally verified form.
• To extend prior classifications by handling both abelian and non-abelian nilradicals in a unified framework.

Proposed method

• Utilize the automorphism group of the nilradical to simplify the structure equations of the full Lie algebra to a canonical form.
• Apply real Jordan canonical forms to classify the action of derivations on the nilradical.
• Use one-parameter subgroups of the automorphism group to generate transformations that simplify matrix representations of derivations.
• Reduce the derivation matrix of the complement to canonical form by exploiting symplectic and orthogonal invariance under the action of the semisimple part.
• Employ Maple to compute a complete basis of the derivation algebra and exponentiate to obtain generating automorphisms.
• Classify algebras by analyzing the canonical forms of the adjoint action of the complement on the nilradical, leading to distinct parent cases based on eigenvalue and block structure.

Experimental results

Research questions

• RQ1 What are all the isomorphism classes of real indecomposable solvable Lie algebras of dimension 2 through 7 with a nilradical of codimension one?
• RQ2 How can the structure constants of such Lie algebras be simplified using automorphisms of the nilradical?
• RQ3 What canonical forms arise for the derivation matrices acting on the nilradical, and how do they classify the algebras?
• RQ4 How do the real Jordan and symplectic canonical forms of the derivation matrices determine the isomorphism types?
• RQ5 What are the distinct structural types that emerge when the semisimple part of the derivation algebra is isomorphic to sl(2,R), sp(4,R), or so(3,1,R)?

Key findings

• The classification yields a complete list of isomorphism classes for all real indecomposable solvable Lie algebras of dimension 2 through 7 with codimension one nilradicals.
• For each nilradical, the derivation algebra's structure determines the isomorphism class, with three main parent cases arising from different canonical forms of the derivation matrix.
• The automorphism group of the nilradical, particularly its semisimple part isomorphic to h(J2), enables the reduction of the derivation matrix to a minimal canonical form.
• The classification includes both abelian and non-abelian nilradicals, with detailed treatment of cases where the semisimple part is isomorphic to sl(2,R), sp(4,R), or so(3,1,R).
• Explicit multiplication tables for all isomorphism classes are provided in Appendix B, confirming the completeness of the classification.
• The method successfully reduces complex structure constants to canonical forms through systematic use of automorphisms and matrix canonicalization, enabling full enumeration.