AI 中文总结
研究椭圆量子代数\(\mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_N)\)，构造中心元素\(\mathfrak{z}(z)\)并证明其可表为量子行列式，得到刘维尔公式椭圆类似物，还建立了如雅可比比例定理和西尔维斯特定理等行列式恒等式。
AI 中文摘要
椭圆量子代数是由杨-巴克斯特方程的椭圆解所表征的代数结构。本文为椭圆量子代数\(\mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_{N})\)构造了一族中心元素\(\mathfrak{z}(z)\)，证明其可表示为量子行列式，得到刘维尔公式的椭圆类似物。此外，还建立了行列式恒等式，包括雅可比比例定理和西尔维斯特定理。
英文摘要
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.
