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arXiv · 2412.19385

Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$

Naihuan Jing·Li Zheng·Jian Zhang
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Abstract

We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\mathfrak{gl}}_{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of $\mathrm{U}_q(\widehat{\gl}_{M|N})$.

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Naihuan Jing, Li Zheng, Jian Zhang. 2024-12-26. Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$. https://doi.org/10.1007/s11005-025-01966-5

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