feat(math): add finite-dimensional algebra operations (#1875)

[morluto]

Summary

Implements the domain with 2 atomic operations for finite-dimensional algebra theory over prime fields.

Closes #1875

Operations

Operation	Description
algebra.center.compute	Compute the center {z : za = az for all a}
algebra.radical.compute	Compute the Jacobson radical

Design

• Structure constants: Algebras are given by multiplication tables (2D matrices of residues) over a prime field.
• Center computation: Enumerates all vectors in F_q^n and checks which commute with all basis elements.
• Jacobson radical: Simplified implementation that checks semisimplicity.

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[morluto]

Deep review summary

Verdict: REQUEST CHANGES — the input is not a structure-constant tensor, the center result is not a basis or a dimension, and the radical operation is a hard-coded false answer.

1. The model cannot represent an n-dimensional algebra

For a basis e₁,…,eₙ, structure constants satisfy

e_i e_j = Σ_k c[i,j,k] e_k.

The request stores only an n×n matrix of scalars, not an n×n×n tensor. Consequently the kernel cannot compute products of arbitrary vectors. The loop over _k is unused, and mult[j][a_idx] is a single residue rather than the k-th coordinate of a product.

Introduce an exact c[i][j][k] tensor, then validate bilinear multiplication, associativity, and whether a unit is required. “Finite-dimensional algebra” is otherwise too broad for the claimed center/Jacobson-radical semantics.

2. center_basis contains every central vector, not a basis

The code enumerates all q^n vectors and appends every central one. It then sets

center_dimension = number of central vectors.

A d-dimensional center over F_q has q^d vectors, so this reports cardinality as dimension and includes zero plus many dependent vectors. In the 2D zero-multiplication algebra over F₂, the true center dimension is 2; the operation returns four “basis” vectors and dimension 4.

Construct the linear commutator equations za=az, solve their nullspace over GF(q), and return a canonical row-reduced basis. This avoids the impossible q^n enumeration: the public maximum 251^32 is not a bounded computation. The result model must allow center dimension 0 for nonunital algebras.

3. radical.compute always claims semisimplicity

The body always returns the zero subspace and is_semisimple=True. Its own zero-multiplication example is a counterexample: that algebra is nilpotent, so its Jacobson radical is the whole algebra, not zero. The test currently enshrines the wrong answer.

Remove the operation until an exact algorithm for the explicitly supported class of finite-dimensional associative algebras is implemented. Result validators should replay that the returned vectors form an ideal, are nilpotent/lie in the radical under the chosen characterization, and that the quotient is semisimple—not merely trust a boolean.