An algebraic 21-vertex intertwiner for a non-elliptic XYZ Lax operator

Abstract

Fukai and Yamada introduced a Lax operator with a three-dimensional auxiliary space for the spin-1/2 XYZ chain whose entries are algebraic rather than elliptic. We give an explicit algebraic 9 × 9 intertwiner for this operator. The matrix has the 21-vertex pattern and is written directly in terms of the two spectral parameters and the three XYZ couplings. Direct reduction in the radical function field proves the local RLL relation. We also prove an exchange-unitarity identity, generic uniqueness up to scalar, and regularity after a scalar renormalization. Its invariant blocks give a factorization of the characteristic polynomial and show that, on the common finite chart, the rank-loss locus is exactly 1 − R² = 0. We make the constant gauge to the fused eight-vertex matrix explicit and obtain regular XXZ and scaled rational XXX limits. The result supplies the local intertwiner left implicit by the fusion argument; it does not constitute a new integrable model.