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Symmetry algebra generated by Σ^± is sp_{2N} (type C_N)

Prove that the Lie algebra generated by the operators Σ^+ and Σ^- constructed for the periodic Motzkin chain is the rank-N simple Lie algebra of type C_N (the symplectic algebra sp_{2N}), i.e., that one can define Chevalley generators e_i, f_i, h_i satisfying the Serre relations with Cartan matrix A equal to the type-C_N Cartan matrix displayed in the paper.

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Background

After proposing explicit formulas for Σ± and their ladder action, the authors investigate the algebra they generate by constructing derived operators (Σz, Λ±, Λz, etc.) and testing Serre relations.

Based on computations for small N, they conjecture that the resulting symmetry algebra is sp_{2N}, specified by a Cartan matrix of type C with rank equal to the number of sites.

References

Conjecture. The operators Σ± generate algebra (3.6) of rank k=N with A=⎡⎢⎢⎢⎢⎢⎢⎣ 2 −1 0 … 0 0 −1 2 −1 … 0 0 0 −1 2 … 0 0 … … … … … … 0 0 0 … 2 −1 0 0 0 … −1 2 −1 0 0 0 … 0 −2 2 ⎤⎥⎥⎥⎥⎥⎥⎦, i.e., the Lie algebra C_N=𝔰𝔭_{2N}.

— Periodic Motzkin chain: Ground states and symmetries (2504.00835 - Pronko, 1 Apr 2025) in Conjecture, Section 3