[Paper Review] On isomorphism classes and invariants of low dimensional complex filiform Leibniz algebras (part 2)

I. S. Rakhimov, Sharifah Kartini Said Husain|ArXiv.org|Jun 11, 2008

Advanced Topics in Algebra9 references3 citations

TL;DR

This paper presents a complete classification of a subclass of complex filiform Leibniz algebras in dimension 7, using an invariant-based method to determine isomorphism classes. It identifies 23 distinct isomorphism classes and proposes a conjectured formula, $N_n = n^2 - 7n + 15$, for the adapted number of isomorphism classes in dimension $n$, confirmed computationally for $n=9$ using Maple 10.

ABSTRACT

The paper is an implementation in low dimensional cases of the classification method presented before by Rakhimov and Bekbaev. We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from the naturally graded non-Lie filiform Leibniz algebras. A hypothetic formula for the adapted number of isomorphism classes is given.

Motivation & Objective

• To classify a specific subclass of non-Lie complex filiform Leibniz algebras in low dimensions.
• To apply an invariant-based method for distinguishing isomorphism classes of Leibniz algebras.
• To provide a complete list of isomorphism classes for 7-dimensional algebras in the second class of naturally graded non-Lie filiform Leibniz algebras.
• To conjecture a general formula for the number of isomorphism classes in higher dimensions.
• To validate the formula computationally using Maple 10 for dimension 9.

Proposed method

• The classification is based on structural constants derived from the Leibniz identity and adapted basis conditions.
• The method uses invariants to distinguish isomorphism classes, avoiding brute-force orbit analysis under $GL_n(\mathbb{C})$.
• Algebras are parametrized by coefficients $\alpha_i, \beta_i, \theta, \gamma$, and isomorphism classes are determined by analyzing these parameters under equivalence relations.
• The approach splits the algebra family into subsets based on non-zero parameter patterns, enabling systematic classification.
• A computer algebra system (Maple 10) is used to verify invariance of expressions under isomorphism transformations.
• The adapted number of isomorphism classes is computed by enumerating distinct parameter families up to isomorphism.

Experimental results

Research questions

• RQ1 How many isomorphism classes exist for 7-dimensional complex filiform Leibniz algebras in the second class of naturally graded non-Lie filiform algebras?
• RQ2 Can a general formula be derived for the number of isomorphism classes of $n$-dimensional non-Lie complex filiform Leibniz algebras?
• RQ3 Are the structural invariants sufficient to fully distinguish isomorphism classes in this subclass?
• RQ4 How can computational tools like Maple 10 be effectively used to verify isomorphism invariants in parametric families?
• RQ5 Does the proposed formula $N_n = n^2 - 7n + 15$ hold beyond dimension 7, particularly in dimension 9?

Key findings

• A complete classification of 23 isomorphism classes is established for 7-dimensional complex filiform Leibniz algebras in the second class of naturally graded non-Lie algebras.
• The adapted number of isomorphism classes for dimension 7 is confirmed to be $N_8 = 23$, corresponding to the 23 distinct algebras listed.
• The conjectured formula $N_n = n^2 - 7n + 15$ is validated for dimension 9 through computational verification.
• The parametric families of algebras are fully classified by analyzing the invariants of their structure constants under isomorphism.
• The use of Maple 10 enables efficient verification of invariance conditions for complex parametric expressions.
• The classification method successfully distinguishes isomorphism classes without requiring full orbit enumeration under $GL_n(\mathbb{C})$.