Central Elements and Determinantal Identities in the Elliptic Quantum Algebra $ \mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_N)$

Yingjie Hu; Zheng Li; Jian Zhang

Central Elements and Determinantal Identities in the Elliptic Quantum Algebra $ \mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_N)$

Quantum Algebra 2026-07-13 v1

Authors: Yingjie Hu , Zheng Li , Jian Zhang

Abstract

Elliptic quantum algebra is the algebraic structure characterized by the elliptic solution of the Yang-Baxter equation. In this paper, we construct a family of central elements $\mathfrak{z}(z)$ for the elliptic quantum algebra $\mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_{N})$ and show that they can be expressed as quantum determinants, yielding an elliptic analogue of the Liouville formula. In addition, we establish determinantal identities, including Jacobi's ratio theorem and Sylvester's theorem.

Keywords

elliptic quantum algebra central elements quantum determinants Liouville formula Jacobi's ratio theorem Sylvester's theorem

Cite

@article{arxiv.2607.11455,
  title  = {Central Elements and Determinantal Identities in the Elliptic Quantum Algebra \( \mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_N)\)},
  author = {Yingjie Hu and Zheng Li and Jian Zhang},
  journal= {arXiv preprint arXiv:2607.11455},
  year   = {2026}
}