# Analogues of Lusztig's higher order relations for the q-Onsager algebra

Abstract : Let $A,A^*$ be the generators of the $q-$Onsager algebra. Analogues of Lusztig's $r-th$ higher order relations are proposed. In a first part, based on the properties of tridiagonal pairs of $q-$Racah type which satisfy the defining relations of the $q-$Onsager algebra, higher order relations are derived for $r$ generic. The coefficients entering in the relations are determined from a two-variable polynomial generating function. In a second part, it is conjectured that $A,A^*$ satisfy the higher order relations previously obtained. The conjecture is proven for $r=2,3$. For $r$ generic, using an inductive argument recursive formulae for the coefficients are derived. The conjecture is checked for several values of $r\geq 4$. Consequences for coideal subalgebras and integrable systems with boundaries at $q$ a root of unity are pointed out.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-00925248
Contributor : Pascal Baseilhac Connect in order to contact the contributor
Submitted on : Tuesday, January 7, 2014 - 5:28:59 PM
Last modification on : Tuesday, January 11, 2022 - 5:56:07 PM

### Identifiers

• HAL Id : hal-00925248, version 1
• ARXIV : 1312.3433

### Citation

P. Baseilhac, T. T. Vu. Analogues of Lusztig's higher order relations for the q-Onsager algebra. 2013. ⟨hal-00925248⟩

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