sexpansion 1.0.0

S-expansions of Lie algebras with finite abelian semigroups

sexpansion

S-expansions of Lie algebras with finite abelian semigroups, in Python.

This package is a full port of the Sexpansion Java library described in:

C. Inostroza, I. Kondrashuk, N. Merino, F. Nadal, A Java Library to Perform S-Expansions of Lie Algebras, Axioms (2025).

The S-expansion method combines a Lie algebra G with a finite abelian semigroup S to produce new, generally non-isomorphic Lie algebras G_S = S ⊗ G, from which smaller algebras can be extracted: resonant subalgebras, 0_S-reduced algebras, and 0_S-reductions of resonant subalgebras.

Installation

pip install sexpansion

Requires Python ≥ 3.10 and NumPy. The catalogue of all non-isomorphic semigroups of orders 2–6 (17,281 tables) ships with the package.

Quickstart

Expand sl(2, R) with the semigroup S⁹⁹¹₍₅₎, extract the resonant subalgebra, and 0_S-reduce it (the paper's Section 5.5 example):

from sexpansion import LieAlgebra, Resonance, load_semigroup

sl2 = LieAlgebra.sl2()                 # [X1,X2]=-2X3, [X1,X3]=2X2, [X2,X3]=2X1
s991 = load_semigroup(5, 991)          # from the bundled catalogue

expanded = sl2.s_expand(s991)          # G_S = S (x) G, 15 generators
resonant = expanded.resonant_subalgebra(
    Resonance(s0=(0, 1, 2), s1=(0, 3, 4)),   # S = S0 u S1
    v0=(0,), v1=(1, 2),                       # G = V0 (+) V1
)
final = resonant.zero_reduced()        # 6 generators: su(2) (+) sl(2,R)

print(final.det())                     # != 0 -> semi-simplicity preserved
print(final.signature())               # (2, 4, 0)

Work with the semigroup catalogue:

from sexpansion import load_semigroups, find_all_resonances

for sg in load_semigroups(4):
    if sg.is_commutative and (zero := sg.find_zero()) is not None:
        print(sg.sem_id, zero, len(find_all_resonances(sg)))

Readable reports for the physicist's eye (pass one_based=True to match the λ₁…λₙ labelling of the paper):

from sexpansion.reports import commutator_table
print(commutator_table(final, one_based=True))
# [X_(1,2), X_(2,4)] = -2 X_(3,5)
# ...

Conventions

• Everything is 0-based: semigroup elements are 0..n-1 and Lie algebra generators 0..n-1. The catalogue files (sem.2…sem.6) use 1-based labels and are converted on load; the paper and the Java library print 1-based labels, which the report functions reproduce with one_based=True.
• Semigroup.table[i, j] is the product i * j.
• LieAlgebra.constants[i, j, k] is C_ij^k in [X_i, X_j] = C_ij^k X_k; set_constant fills the antisymmetric partner automatically.
• ExpandedAlgebra.tensor[i, a, j, b, k, c] is C_(i,a)(j,b)^(k,c) = K_ab^c C_ij^k.
• Metrics are in generator-major order: row i * m + a ↔ generator X_(i,a).

Mapping from the Java library

Java	Python
Semigroup.isAssociative() / isCommutative()	Semigroup.is_associative / is_commutative
Semigroup.findZero() → -1	Semigroup.find_zero() → None
Semigroup.loadFromFile(...)	load_semigroups(order) / load_all_semigroups()
Semigroup.isResonant(S0, S1)	is_resonant(sg, s0, s1)
isResonatF / findAllResonancesF/F2	filtered=True flag
Semigroup.permuteWith(SetS) / permute()	Semigroup.permute(sigma) / all_images()
Semigroup.isotest(B)	Semigroup.isomorphism_test(other)
SetS	tuple[int, ...], itertools, Permutation
StructureConstantSet	LieAlgebra
getExpandedStructureConstant(s)	LieAlgebra.s_expand(semigroup)
StructureConstantSetExpanded{,Reduced,Resonant,ResonantReduced}	ExpandedAlgebra + .resonant_subalgebra(...) / .zero_reduced()
cartanKillingMetric() / ...Pretty()	cartan_killing_metric(restricted=False) / default
show* methods	sexpansion.reports functions (return strings)
Jama Matrix	NumPy arrays (np.linalg.det/eigh/inv)

Deliberate deviations from the Java code: the anti-isomorphism branch of isotest (which could never trigger) is fixed; the non-terminating maximalAbelianSubalgebra is replaced by a correct algorithm; the isResonatF typo is renamed.

Examples and tests

The examples/ folder contains ten Jupyter notebooks porting representative programs from the paper (Appendix A), from associativity checks up to the full S-expansion pipeline. They are committed with executed outputs, so the results can be read directly on GitHub without installing anything; to run them yourself, pip install sexpansion jupyterlab and open the notebooks (re-execute in order — later cells reuse earlier ones).

The test suite reproduces the paper's Table 5 catalogue statistics (e.g. order 6: 2,059 semigroups with 25,512 resonances) and, when the Java repository is present, cross-checks against its captured outputs (SEXPANSION_JAVA_OUTPUTS environment variable; run the slow catalogue scans with pytest -m slow).

License

GPL-3.0-only, matching the original Java library. If you use this package in academic work, please cite the paper above.