AI 中文总结
该研究解决了q - 昂萨格代数PBW基与中心化子的五个猜想，通过证明特定根向量给出PBW基、建立多个PBW基及确定相关子代数中心化子，运用多种论证方法得出四个单族交替多项式子代数是极大交换的结论。
AI 中文摘要
我们解决了关于q - 昂萨格代数的PBW基和中心化子的五个猜想。首先证明当q不是单位根时，Baseilhac - Kolb根向量在每个线性序下给出一个PBW基，去除了先前的超越假设。接着在交替生成元中建立了十二个PBW基，并表明它们在交替中心扩张的任意标量中心特化下持续存在。最后确定了负和虚交替子代数的中心化子，并表明所有四个单族交替多项式子代数是极大交换的。证明结合了Damiani根向量的显式拉直、退化\(\mathrm{gr} O_q \simeq U_q^+(\widehat{\mathfrak{sl}}_2)\)以及大指标三角交叉论证。
英文摘要
We prove that, over an arbitrary field and whenever $q$ is not a root of unity, the Baseilhac--Kolb root vectors form a PBW basis of the $q$-Onsager algebra for every total order on the positive roots of $\widehat{\mathfrak{sl}}_2$. This removes the previous transcendence hypothesis. We establish twelve PBW bases in the alternating generators and show that they persist under arbitrary scalar central specialisation of the alternating central extension. We determine the centraliser of the negative alternating subalgebra and that of the first imaginary alternating generator, and deduce that the four single-family alternating polynomial subalgebras are maximal commutative. Together, these results settle four conjectures of Terwilliger and, in characteristic different from $2$, a conjecture of Baseilhac and Belliard.
Comments36 pages
