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Symbolic computation of optimal systems of subalgebras of three- and four-dimensional real Lie algebras

Luca Amata, Francesco Oliveri, Emanuele Sgroi

TL;DR

This work introduces a $p$-family based, algorithmic framework for computing optimal systems of subalgebras of finite-dimensional real Lie algebras, implemented in the Wolfram Mathematica package SymboLie. By leveraging inner automorphisms to define a preorder and encoding subalgebra families in directed graphs, the method automates the determination of inequivalent subalgebras without solving general quadratic systems. The authors apply the framework to all 3- and 4-dimensional real Lie algebras classified by Patera and Winternitz (PW), finding results that largely agree with PW’s 1977 classifications, with two noted exceptions arising from the $p$-family definition. The approach provides a fast, automated, and scalable tool for constructing optimal systems and offers insight into equivalences among subalgebras, potentially facilitating extensions to higher dimensions.

Abstract

The complete optimal systems of subalgebras of all nonisomorphic three- and four-dimensional real Lie algebras are analyzed by the program \symbolie running in the computer algebra system \emph{Wolfram Mathematica}\texttrademark. The approach uses the definition of $p$-families of Lie subalgebras whose set can be partitioned by introducing a binary relation (reflexive and transitive, though not necessarily symmetric) induced by inner automorphisms of the Lie algebra. The results, produced in a few minutes by \symbolie, represent a good test for the program; in fact, except for minor differences that are discussed, the results confirm those given in 1977 in a paper by Patera and Winternitz.

Symbolic computation of optimal systems of subalgebras of three- and four-dimensional real Lie algebras

TL;DR

This work introduces a -family based, algorithmic framework for computing optimal systems of subalgebras of finite-dimensional real Lie algebras, implemented in the Wolfram Mathematica package SymboLie. By leveraging inner automorphisms to define a preorder and encoding subalgebra families in directed graphs, the method automates the determination of inequivalent subalgebras without solving general quadratic systems. The authors apply the framework to all 3- and 4-dimensional real Lie algebras classified by Patera and Winternitz (PW), finding results that largely agree with PW’s 1977 classifications, with two noted exceptions arising from the -family definition. The approach provides a fast, automated, and scalable tool for constructing optimal systems and offers insight into equivalences among subalgebras, potentially facilitating extensions to higher dimensions.

Abstract

The complete optimal systems of subalgebras of all nonisomorphic three- and four-dimensional real Lie algebras are analyzed by the program \symbolie running in the computer algebra system \emph{Wolfram Mathematica}\texttrademark. The approach uses the definition of -families of Lie subalgebras whose set can be partitioned by introducing a binary relation (reflexive and transitive, though not necessarily symmetric) induced by inner automorphisms of the Lie algebra. The results, produced in a few minutes by \symbolie, represent a good test for the program; in fact, except for minor differences that are discussed, the results confirm those given in 1977 in a paper by Patera and Winternitz.

Paper Structure

This paper contains 8 sections, 77 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical preliminaries and notation
  3. One-dimensional Lie subalgebras
  4. Higher dimensional Lie subalgebras
  5. How to use SymboLie
  6. Analysis of three-dimensional real Lie algebras
  7. Analysis of four-dimensional Lie algebras
  8. Conclusions

Figures (7)

  • Figure 1: One-dimensional optimal system for the Lie algebra $A_{3,8}$ computed by SymboLie. $1\to \{\Xi_1\}$, $2\to \{\Xi_2\}$, $3\to \{\Xi_3\}$, $4\to \{\Xi_1+a_1 \Xi_2\}$, $5\to \{\Xi_1+a_1 \Xi_3\}$, $6\to \{\Xi_2+a_1 \Xi_3\}$, $7\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_3\}$.
  • Figure 2: One-dimensional optimal system for the Lie algebra $2A_2$ computed by SymboLie. $1\to \{\Xi_1\}$, $2\to \{\Xi_2\}$,$3\to \{\Xi_3\}$,$4\to \{\Xi_4\}$,$5\to \{\Xi_1+a_1\Xi_2\}$,$6\to \{\Xi_1+a_1 \Xi_3\}$,$7\to \{\Xi_2+a_1 \Xi_3\}$,$8\to \{\Xi_1+a_1 \Xi_4\}$,$9\to \{\Xi_2+a_1 \Xi_4\}$,$10\to \{\Xi_3+a_1 \Xi_4\}$,$11\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_3\}$,$12\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_4\}$,$13\to \{\Xi_1+a_1 \Xi_3+a_2 \Xi_4\}$,$14\to \{\Xi_2+a_1 \Xi_3+a_2 \Xi_4\}$,$15\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_3+a_3 \Xi_4\}$
  • Figure 3: Two-dimensional optimal system for the Lie algebra $A_{3,6}\oplus A_1$ computed by SymboLie. $1\to \{\Xi_1,\Xi_2\}$, $2\to \{\Xi_1,\Xi_4\}$, $3\to \{\Xi_2,\Xi_4\}$, $4\to \{\Xi_3,\Xi_4\}$, $5\to \{\Xi_1, \Xi_2+a_1 \Xi_4\}$, $6\to \{\Xi_1+a_1 \Xi_2,\Xi_4\}$, $7\to \{\Xi_1+a_1 \Xi_3,\Xi_4\}$, $8\to \{\Xi_2+a_1 \Xi_3,\Xi_4\}$, $9\to \{\Xi_1+a_1 \Xi_4,\Xi_2\}$, $10\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_3,\Xi_4\}$, $11\to \{\Xi_1+a_1 \Xi_4,a_2 \Xi_2+a_3 \Xi_4\}$
  • Figure 4: Two-dimensional optimal system for the Lie algebra $A_{3,7}^a \oplus A_1$ computed by SymboLie. $1\to \{\Xi_1,\Xi_2\}$, $2\to \{\Xi_1,\Xi_4\}$, $3\to \{\Xi_2,\Xi_4\}$, $4\to \{\Xi_3,\Xi_4\}$, $5\to \{\Xi_1,\Xi_2+a_1 \Xi_4\}$, $6\to \{\Xi_1+a_1 \Xi_2,\Xi_4\}$, $7\to \{\Xi_1+a_1 \Xi_3,\Xi_4\}$, $8\to \{\Xi_2+a_1 \Xi_3,\Xi_4\}$, $9\to \{\Xi_1+a_1 \Xi_4,\Xi_2\}$, $10\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_3,\Xi_4\}$, $11\to \{\Xi_1+a_1 \Xi_4,\Xi_2+a_2 \Xi_4\}$
  • Figure 5: One-dimensional optimal system for the Lie algebra $A_{4,5}^{a,b}$ computed by SymboLie. $1\to \{\Xi_1\}$, $2\to \{\Xi_2\}$,$3\to \{\Xi_3\}$,$4\to \{\Xi_4\}$,$5\to \{\Xi_1+a_1\Xi_2\}$,$6\to \{\Xi_1+a_1 \Xi_3\}$,$7\to \{\Xi_2+a_1 \Xi_3\}$,$8\to \{\Xi_1+a_1 \Xi_4\}$,$9\to \{\Xi_2+a_1 \Xi_4\}$,$10\to \{\Xi_3+a_1 \Xi_4\}$,$11\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_3\}$,$12\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_4\}$,$13\to \{\Xi_1+a_1 \Xi_3+a_2 \Xi_4\}$,$14\to \{\Xi_2+a_1 \Xi_3+a_2 \Xi_4\}$,$15\to \{\Xi_1+a_1 \Xi_2+a_2 \Xi_3+a_3 \Xi_4\}$
  • ...and 2 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3
  • Example 2
  • Definition 4
  • Remark 1
  • Definition 5
  • Remark 2
  • Remark 3
  • ...and 4 more