A Quiver-Algebraic Construction of the Standard Model Gauge Algebra

The Standard Model gauge algebra su(3) ⊕ su(2) ⊕ u(1) is recovered as the Cartan-Killing image of a symmetric bilinear form on the first homology of a torus, where the torus is the product of the cycle space of a four-vertex quiver and an independent temporal circle. The quiver has four vertices, six directed edges, first Betti number three, and one vertex of in-degree three. A symmetric bilinear form B on the rank-three cycle space, constructed in Stewart (2026e) from a signed-edge partition of the convergence vertex's exit, has the Cartan matrix of A₂ × A₁ as its matrix in the cycle-generator basis. Cartan-Killing on A₂ × A₁ returns the rank-three semisimple algebra su(3) ⊕ su(2). The rank-four extension required for the full Standard Model gauge algebra is supplied by a fourth generator γ_τ identified with the fundamental class of the temporal circle of the torus traced by the iterated path algebra endomorphism. The generator γ_τ is independent of the cycle space because the temporal circle is independent of the quiver's first homology, fails the involution relation γ² = e satisfied by the spatial generators because S¹ has fundamental group ℤ, and commutes with all spatial generators because temporal and spatial coordinates of the torus are independent. The failure of the involution forces the abelian character of the fourth factor. The bilinear form extends to a 4 × 4 block-diagonal matrix B̃ with diagonal entry B̃(γ_τ, γ_τ) = 4 derived from the dual-axis structure of the quiver's boundary vertex pair. The Cartan-Killing image of B̃ is su(3) ⊕ su(2) ⊕ u(1). The structural invariant k₁ = B̃(γ_τ, γ_τ) / B̃(spatial diagonal) = 4 / 2 = 2. The Weinberg angle sin²θ_W = 1/3 at the fundamental scale follows from k₁ = 2 by a coupling-strength correspondence between bilinear-form diagonal and gauge-coupling reciprocal. The 3+1 dimensional structure follows from the rank-three spatial cycle basis plus the rank-one temporal generator, with the involution failing only on the temporal generator. The renormalization-group running of sin²θ_W from the fundamental scale to the electroweak scale is named as an open sub-problem. **Keywords,** path algebra, quiver representation, Cartan matrix, A₂ × A₁, Standard Model gauge algebra, Standard Model gauge group, Cartan-Killing correspondence, symmetric bilinear form, first homology, torus, root system, gauge coupling, hypercharge, Weinberg angle, 3+1 dimensions