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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 ≥ 4. Consequences for coideal subalgebras and integrable systems with boundaries at q a root of unity are pointed out.

© 2014 AIP Publishing LLC

Received 09 April 2014
Accepted 20 July 2014
Published online 14 August 2014

Acknowledgments:
We are indebted to P. Terwilliger for a careful reading of the first version of the paper, and sharing with us some of the results presented in Sec. II. P.B thanks S. Baseilhac and S. Kolb for discussions.

Article outline:

I. INTRODUCTION
II. HIGHER ORDER RELATIONS FROM THE THEORY OF TRIDIAGONAL PAIRS
A. Tridiagonal pairs of *q*-Racah type
B. Tridiagonal systems
C. Higher order tridiagonal relations
D. Higher order *q*-Dolan-Grady relations
III. HIGHER ORDER RELATIONS FOR THE *Q*-ONSAGER ALGEBRA: RECURSION FOR GENERATING THE COEFFICIENTS
A. Proof of the relations for *r* = 2
B. Proof of the relations for *r* = 3
C. Relations for *r* generic
D. Comments
IV. CONCLUDING REMARKS