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

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.

💡 Research Summary

The paper presents a systematic, computer‑assisted method for constructing optimal systems of subalgebras of all non‑isomorphic three‑ and four‑dimensional real Lie algebras. The authors have implemented the method in a Mathematica package called SymbolicLie (stylised as SymboLie in the text). The motivation stems from the classical problem of classifying inequivalent Lie subgroups of a symmetry group of a differential equation; inequivalent subgroups correspond to inequivalent Lie subalgebras, and an optimal system is a minimal set of representatives such that every subalgebra is conjugate to one of them under inner automorphisms.

Theoretical framework.
Starting from a basis ({\Xi_i}{i=1}^r) of a real Lie algebra (L) with structure constants (C^\gamma{\alpha\beta}), the authors rewrite the Lie bracket in coordinate form and observe that a (d)-dimensional subalgebra is determined by a rank‑(d) matrix (h_{\alpha i}) satisfying a system of quadratic equations (eq. 4). For one‑dimensional subalgebras this reduces to a vector of coefficients (\mathbf{f}\in\mathbb{R}^r); inner automorphisms act linearly on this vector.

To avoid the ad‑hoc, case‑by‑case simplifications traditionally used, the authors introduce (p)-families. A (p)-family is a collection of one‑dimensional subalgebras described by a binary mask (s\in{0,1}^r\setminus{0}) and a tuple of functions ((f_1,\dots,f_r)). The integer (p=\sum_i s_i) counts the number of active basis elements, and the Jacobian of the active components must have rank (p). This definition captures the functional independence of the coefficients and provides a combinatorial handle on the multitude of possible subalgebras.

Relation (R) and preorder structure.
Two (p)-families (X) and (Y) are related by (X,R,Y) if there exists an inner automorphism that maps (X) to a family whose integer label (c) (derived from the binary mask) coincides with that of (Y). The relation is reflexive and transitive but not necessarily symmetric, thus forming a preorder rather than an equivalence relation. The authors encode the preorder as a directed multigraph (G(L)) whose vertices are the (p)-families and edges correspond to automorphisms effecting the relation. The adjacency matrix of this graph is used to compute transitive closures efficiently.

A short‑lexicographic ordering (slex) on the binary masks together with the integer label (c) provides a deterministic way to select a minimal representative from each equivalence class. The algorithm proceeds by constructing the graph, performing a topological‑like reduction according to slex, and extracting the set (\Theta_{\mathrm{Int}(L)}(L)) of representatives, i.e., the optimal system.

Higher‑dimensional subalgebras.
The notion of a (p)-family is extended to (d)-dimensional subalgebras ((1<d<r)). A collection ({X_k}{k=1}^d) forms a (p)-family of dimension (d) if each (X_k) is a (p_k)-family of one‑dimensional subalgebras, the sum (p=\sum_k p_k) equals the total number of independent parameters, the coefficient matrix (|f^\alpha_k|) has rank (d), and the closure condition (eq. 5) holds for some constants (\lambda^k{ij}). By fixing the basis of each subalgebra in reduced row‑echelon form, the authors eliminate redundancy and guarantee a unique matrix representation for each family.

Implementation in SymbolicLie.
The Mathematica package accepts the structure constants of a Lie algebra, automatically generates the inner automorphism group (via exponentials of the adjoint representation), enumerates all possible binary masks, builds the set of (p)-families, constructs the preorder graph, and finally extracts the optimal system. The user can specify the desired subalgebra dimension and optionally restrict the set of automorphisms. The package is freely available at the authors’ university site.

Computational results.
The authors applied SymbolicLie to the full list of non‑isomorphic real Lie algebras of dimension three (nine algebras) and four (fifteen algebras) classified by Patera and Winternitz (1977). For each algebra they computed optimal systems of one‑, two‑ (and three‑ for the 4‑dimensional case) dimensional subalgebras. The entire computation for a given algebra took from a few seconds up to a couple of minutes on a standard workstation, demonstrating the practicality of the approach.

Comparison with classical results.
The optimal systems produced by SymbolicLie were compared with the tables in Patera‑Winternitz. In the three‑dimensional case there was complete agreement. In the four‑dimensional case a handful of discrepancies appeared, all of which were traced back to the non‑symmetry of the preorder relation (R). For example, in the algebra (L_4) with non‑zero brackets (