A conjecture concerning the q-Onsager algebra

Abstract The q-Onsager algebra O q is defined by two generators W 0 , W 1 and two relations called the q-Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for O q with elements denoted { B n δ + α 0 } n = 0 ∞ , { B n δ + α 1 } n = 0 ∞ , { B n δ } n = 1 ∞ . In their recent study of a current algebra A q , Baseilhac and Belliard conjecture that there exist elements { W − k } k = 0 ∞ , { W k + 1 } k = 0 ∞ , { G k + 1 } k = 0 ∞ , { G ˜ k + 1 } k = 0 ∞ in O q that satisfy the defining relations for A q . In order to establish this conjecture, it is desirable to know how the elements on the second displayed line above are related to the elements on the first displayed line above. In the present paper, we conjecture the precise relationship and give some supporting evidence. This evidence consists of some computer checks on SageMath due to Travis Scrimshaw, a proof of the analog conjecture for the Onsager algebra O , and a proof of our conjecture for a homomorphic image of O q called the universal Askey-Wilson algebra.

[1]  P. Baseilhac,et al.  Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory , 2007, hep-th/0703106.

[2]  A new (in)finite-dimensional algebra for quantum integrable models , 2005, math-ph/0503036.

[3]  P. Baseilhac,et al.  Form factors of the half-infinite XXZ spin chain with a triangular boundary , 2014 .

[4]  P. Baseilhac,et al.  Correlation functions of the half-infinite XXZ spin chain with a triangular boundary , 2013, 1309.7785.

[5]  K. Shigechi,et al.  A New Current Algebra and the Reflection Equation , 2009, 0906.1482.

[6]  Paul M. Terwilliger,et al.  The augmented tridiagonal algebra , 2009, 0904.2889.

[7]  P. Baseilhac,et al.  BRAID GROUP ACTION AND ROOT VECTORS FOR THE q-ONSAGER ALGEBRA , 2017, Transformation Groups.

[8]  Michael Grady,et al.  Conserved charges from self-duality , 1982 .

[9]  P. Baseilhac,et al.  The half-infinite XXZ chain in Onsagerʼs approach , 2012, 1211.6304.

[10]  P. Baseilhac,et al.  Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators , 2017, 1707.04574.

[11]  E. Sklyanin.  Boundary conditions for integrable quantum systems , 1988 .

[12]  S. Kolb.  Quantum symmetric Kac–Moody pairs , 2012, 1207.6036.

[13]  P. Baseilhac,et al.  A deformed analogue of Onsager’s symmetry in the XXZ open spin chain , 2005, hep-th/0507053.

[14]  Weiqiang Wang,et al.  A Drinfeld type presentation of affine $\imath$quantum groups I: split ADE type , 2020, 2009.04542.

[15]  P. Baseilhac,et al.  Non-Abelian symmetries of the half-infinite XXZ spin chain , 2016, 1611.05390.

[16]  TerwilligerPaul.  The Subconstituent Algebra of an Association Scheme (Part III) , 1993 .

[17]  $\imath$Hall algebra of the projective line and $q$-Onsager algebra , 2020, 2010.00646.

[18]  Tatsuro Ito,et al.  Some algebra related to P- and Q-polynomial association schemes , 1999, Codes and Association Schemes.

[19]  B. Davies.  Onsager’s algebra and the Dolan–Grady condition in the non‐self‐dual case , 1991 .

[20]  Tatsuro Ito,et al.  Tridiagonal pairs of q-Racah type , 2008, 0807.0271.

[21]  Paul M. Terwilliger.  The q-Onsager Algebra and the Universal Askey-Wilson Algebra , 2018, 1801.06083.

[22]  Paul Terwilliger.  Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations , 2001 .

[23]  Deformed Dolan-Grady relations in quantum integrable models , 2004, hep-th/0404149.

[24]  P. Baseilhac,et al.  An attractive basis for the q−Onsager algebra , 2017, 1704.02950.

[25]  J. Perk.  The early history of the integrable chiral Potts model and the odd–even problem , 2015, 1511.08526.

[26]  B. Davies.  Onsager's algebra and superintegrability , 1990 .

[27]  Paul M. Terwilliger.  The Universal Askey-Wilson Algebra , 2011, 1104.2813.

[28]  P. Baseilhac,et al.  Generalized q-Onsager Algebras and Boundary Affine Toda Field Theories , 2009, 0906.1215.

[29]  N. Crampé,et al.  FRT presentation of classical Askey–Wilson algebras , 2018, Letters in Mathematical Physics.

[30]  An integrable structure related with tridiagonal algebras , 2004, math-ph/0408025.

[31]  Paul M. Terwilliger,et al.  The Subconstituent Algebra of an Association Scheme (Part III) , 1993 .

[32]  Paul M. Terwilliger.  An action of the free product Z2⋆Z2⋆Z2 on the q-Onsager algebra and its current algebra , 2018, Nuclear Physics B.

[33]  Paul M. Terwilliger.  The q-Onsager algebra and the positive part of Uq(slˆ2) , 2017 .

[34]  N. Crampé,et al.  FRT presentation of the Onsager algebras , 2017, 1709.08555.

[35]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .