On the algebras Uq±(AN) : from a constructive-computational viewpoint

Abstract

Let $U_q^{+}(A_N)$ (resp. $U_q^{-}(A_N)$) be the $(+)$-part (resp. $(-)$-part) of the Drinfeld-Jimbo quantum group of type A_N over a field K. With respect to Jimbo relations and the PBW K-basis $\mathcal{B}$ of $U_q^{+}(A_N)$ (resp. $U_q^{-}(A_N)$) established by Yamane, it is shown, by constructing an appropriate monomial ordering ${\prec }_{*-d}$ on $\mathcal{B}$, that $U_q^{+}(A_N)$ (resp. $U_q^{-}(A_N)$) is a solvable polynomial algebra. Consequently, further structural properties of $U_q^{+}(A_N)$ (resp. $U_q^{-}(A_N)$) and their modules may be established and realized in a constructive-computational way.

KEYWORD:

• Gröbner basis
• quantum group
• solvable polynomial algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16T20
• 16Z05

Acknowledgments

The author would very much like to thank the anonymous referee for his/her valuable comments, particularly for pointing out an error in constructing a monomial ordering in the proof of Theorem 2.3, which are quite helpful for improving the manuscript. The author is very grateful to professor Huishi Li for proposing the research topic, for very valuable discussions, and for his patient and meticulous guidance during preparing the manuscript.

Additional information

Funding

This project was supported by the National Natural Science Foundation of China (11861061 to Rabigul Tuniyaz).