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An Algorithm for Determining Lie Algebra Types

Tu N. T. C. Nguyen, Tuan A. Nguyen, Vu A. Le

TL;DR

The paper presents an explicit algorithm to determine whether a finite-dimensional complex Lie algebra has Jordan-type, Kronecker-type, or mixed Jordan–Kronecker structure by analyzing the generic pencil of skew-symmetric forms associated to the algebra. It builds on Bolsinov–Zhang Bol's framework, computing the Lie algebra index and the characteristic polynomial p(λ) to classify type, with a Matlab implementation. The method is demonstrated on Vu et al.'s 7-dimensional solvable algebras, yielding clear type assignments (mixed for L1,L2; Kronecker for the others), illustrating practical applicability for structural classification. The work provides a computational tool that links invariant theory of Lie algebras to concrete type classification, aiding systematic analysis and cataloging.

Abstract

This paper investigates the Jordan--Kronecker invariant of finite dimensional complex Lie algebras. We present an explicit algorithm for determining the type of a given Lie algebra from its Jordan--Kronecker invariant. The algorithm is implemented in a specific Matlab program.

An Algorithm for Determining Lie Algebra Types

TL;DR

The paper presents an explicit algorithm to determine whether a finite-dimensional complex Lie algebra has Jordan-type, Kronecker-type, or mixed Jordan–Kronecker structure by analyzing the generic pencil of skew-symmetric forms associated to the algebra. It builds on Bolsinov–Zhang Bol's framework, computing the Lie algebra index and the characteristic polynomial p(λ) to classify type, with a Matlab implementation. The method is demonstrated on Vu et al.'s 7-dimensional solvable algebras, yielding clear type assignments (mixed for L1,L2; Kronecker for the others), illustrating practical applicability for structural classification. The work provides a computational tool that links invariant theory of Lie algebras to concrete type classification, aiding systematic analysis and cataloging.

Abstract

This paper investigates the Jordan--Kronecker invariant of finite dimensional complex Lie algebras. We present an explicit algorithm for determining the type of a given Lie algebra from its Jordan--Kronecker invariant. The algorithm is implemented in a specific Matlab program.

Paper Structure

This paper contains 3 sections, 3 theorems, 7 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Basic definitions and theorems
  2. An algorithm determining Lie algebra types
  3. Application to 7-dimensional Lie algebras

Key Result

Theorem 1.1

Let $\mathcal{A}$ and $\mathcal{B}$ be two skew-symmetric bilinear forms on a complex vector space $V$. Then by an appropriate choice of a basis, their matrices can be simultaneously reduced to the following canonical block-diagonal form: where the pairs of the corresponding blocks $\mathcal{A}_i$ and $\mathcal{B}_i$ can be of the following three types: where $J(\lambda_i)$ denotes the standard J

Theorems & Definitions (10)

  • Theorem 1.1: Bol
  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Proposition 1.2: Bol
  • Definition 1.4: Bol
  • Proposition 2.1
  • proof
  • Example 2.1
  • Example 2.2