Matrix Basis for the Classified Algebra

[AmanieOxana]

Context

get_algebra returns the isomorphism label of the dynamical Lie algebra generated by a set of Pauli strings — one of the four canonical types from Theorem 1 in Aguilar et al. 2024. For example,

generators = p(["XY", "XZ"], n=4)
generators.get_algebra()      # -> "sp(4)"

The label fixes the family (u, so, sp) and the dimension parameter, but it is a name, not a usable mathematical object. Anything algebraic with the result — dimension checks, structure-constant computations, feeding into a Cartan-decomposition pipeline like Wierichs et al. 2025, arXiv:2503.19014 — needs the algebra instantiated as a concrete basis of operators.

This issue is the first step: expose the algebra in defining representation. Cartan involutions, k/p split, and CSAs come in a follow-up.

Task

Add a function that returns the algebra named by get_algebra as a list of matrices in its defining representation, partitioned by direct summand. The output is fully determined by the label, so this is table-driven — no input-dependent computation.

Canonical type	Algebra per summand	Defining rep	# summands
A	so(n+3)	(n+3)×(n+3) real antisymmetric	2^{n_1}
B1	sp(2^{n_2})	2^{n_2+1}×2^{n_2+1} symplectic	2^{n_1}
B2	so(2^{n_2+3})	2^{n_2+3}×2^{n_2+3} real antisymmetric	2^{n_1}
B3	su(2^{n_2+2})	2^{n_2+2}×2^{n_2+2} traceless anti-Hermitian	2^{n_1}

Per summand the basis is a textbook construction:

• so(N): {E_{ij} − E_{ji} : 1 ≤ i < j ≤ N}.
• su(N): a fixed traceless anti-Hermitian basis (generalized Gell-Mann is the obvious choice).
• sp(N): standard basis of anti-Hermitian X with X^T J + J X = 0, with J = [[0, I_N], [-I_N, 0]].

Whatever convention is chosen, it must be fixed and documented — downstream consumers need a stable basis ordering.

The task is to find a basis for the direct sum of these simple Lie algebras.

API sketch

from paulie import get_pauli_string as p

generators = p(["XY", "XZ"], n=4)
basis = generators.get_algebra_basis()
# basis: list of np.ndarray, one per summand,
#   each of shape (dim, N, N) with N the defining-rep size

For a single-summand algebra (n_1 = 0) the list has length 1.

Acceptance criteria

• get_algebra_basis is exposed at the same level as get_algebra.
• For every input where get_algebra succeeds, get_algebra_basis returns the corresponding defining-rep matrix basis
• The algebra basis has the correct dimension and is able to span the lie closure
• Convention (basis ordering, sign choices, choice of J) is documented in the function's docstring.
• Unit tests under tests/
• docs/source/user/classification.rst gets a short example reusing the two examples already on that page.

Out of scope (separate follow-ups)

• Cartan involutions, induced k/p decomposition, and horizontal CSA per family — needed to feed the recursive-CD pipeline of arXiv:2503.19014. Tab. II of that paper specifies the conventions; matching them in the follow-up will minimize friction.

References

• Classification: Aguilar et al., arXiv:2408.00081
• Motivating downstream consumer: Wierichs et al., arXiv:2503.19014, Tab. II
• Existing classifier entry point: paulie.get_algebra

[InderpalSuthar]

I have opened a PR "Add get_algebra_basis for defining-rep matrices of classified DLAs #209", solving this issue . Please review the PR.

[khopade-works]

Hi @AmanieOxana I've opened a PR implementing get_algebra_basis (#210 ).

It turns the label from get_algebra into concrete matrices in the defining representation, returned as a list of stacks one per direct summand (so / su / sp / u). It's exposed at the same level as get_algebra (on both Classification and PauliStringCollection), is fully table-driven (depends only on the label, not the generating Pauli strings), uses fixed/documented conventions, and orders summands deterministically so downstream consumers get a stable basis.

from paulie import get_pauli_string as p

p(["XY", "XZ"], n=4).get_algebra_basis()
# [ndarray (36, 8, 8)]              -> sp(4)
p(["XX", "YY", "ZZ", "ZY"], n=4).get_algebra_basis()
# [ndarray (63, 8, 8), (63, 8, 8)]  -> 2*su(8)

[Adithyaphani]

Hi @AmanieOxana . Took a shot at this — PR here → #212
Main thing I focused on: actually verifying the basis closes under the Lie bracket, not just checking dimensions match. Ran everything locally, 905 tests passing. Happy to iterate based on feedback.

[AmanieOxana]

Hi, we got quite spamed regarding this issue. I must admit that the out of scope comment "An explicit isomorphism between the embedded DLA on ℂ^{2^n} and the defining-rep basis returned here." is misleading, as you will need that for the testing, that you can reach with the basis every element of the closure.
Also I have seen that some of you did not construct a basis for the direct sum, but just stacked basis for the blocks which do not lead to a basis of the direct sum. That is the part you have to put a bit though into and that's why we rated this issue a bit higher. I edited the issue to clear confusion.

[Gmin2]

Hey @AmanieOxana can i start working on this ? or is #212 is near completion?

[AmanieOxana]

Hi @Gmin2 so this is a Hackathon, we look for the best solutions for a problem. #212 is on its way to become a solution but if you have a idea how to make a better implementation you are free to contribute.
Our end goal is to simplify the Cartan decomposition pipeline with paulie and for that a first step is a basis for the DLA, there are multiple embeddings out there. If you find one that fits better the KAK pipeline than #212 happy to discuss it.