Abstract
We define two algebra automorphisms Ͳ0 and Ͳ1 of the q-Onsager algebra {mathcal{B}}_c , which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a PBW basis for {mathcal{B}}_c . We show that the root vectors satisfy q-analogs of Onsager's original commutation relations. The paper is much inspired by I. Damiani's construction and investigation of root vectors for the quantized enveloping algebra of hat{mathfrak{s}{mathfrak{l}}_2} .
Highlights
The Onsager algebra O appeared first in 1944 in L
We show that the root vectors Bγ for γ ∈ R satisfy commutation relations which are q-analogs of the Onsager relations (1.1)
In the following theorem, which is the main result of this paper, we write the real root vectors as Bnδ+α1 for n ∈ Z, and we consider the PBW-basis with respect to the total ordering < defined by (1.6)
Summary
The Onsager algebra O appeared first in 1944 in L. In the following theorem, which is the main result of this paper, we write the real root vectors as Bnδ+α1 for n ∈ Z, and we consider the PBW-basis with respect to the total ordering < defined by (1.6). The commutation relations in Theorem II are again proved by an inductive calculation This calculation is significantly harder than the corresponding calculations in [Dam93] due to the lower order terms in the definition of the imaginary root vectors Bmδ. The term Fn satisfies a recursive formula, the proof of which is deferred to Appendix A
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
