An integrable structure related with tridiagonal algebras

Abstract

The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on the tridiagonal algebraic structure associated with a deformation parameter $q$. Representations are shown to be generated from a class of quadratic algebras, namely the reflection equations. The spectral problem is briefly discussed. Finally, related massive quantum integrable models are shown to be superintegrable.Comment: 11 pages; LaTeX file with amssymb; ; v2: typos corrected, one reference added, to appear in Nucl. Phys.

A deformed analogue of Onsager’s symmetry in the XXZ open spin chain

“…In addition, later on it was argued [18] that this algebra is related with a q−deformed analogue of the Onsager's algebra 5 constructed in [18]. As briefly mentionned in [15,18] this remarkable tridiagonal algebraic structure survives for the most general (non-diagonal) solutions of the reflection equation.…”

“…8 To understand the general case N , it is instructive to look at the special cases N = 1 and N = 2 first [15,18]. For N = 1 in where, for the special case k = 0 we identify…”

“…Expanding in the spectral parameter u, one obtains a pair of relations linear in terms of the elements W −k , W k+1 which ensures the closure of the algebra. For N = 1, they correspond to the Askey-Wilson relations [14,15] [18]. In general, given N a pair of relations of order 2N + 1 in W 0 , W 1 exists, which can be written as a pair of linear recurrence relation of length N + 1 in the fundamental elements W −k , W k+1 , k = 0, 1, ..., N where W −N , W N +1 are nonlinear combinations of lower elements W −k , W k+1 with k = 0, 1, ..., N − 1.…”

“…This might be even more surprising, as the term "boundary quantum group algebra" corresponding to such algebraic structure is used repeatedely but with no reference to proper defining relations independent of the spectral parameter u, derived from the reflection equation 4 . However, a breakthrough recently emerged [14,15] from the study of the algebraic structure encoded in the reflection equation for certain (left purely non-diagonal and right diagonal) boundary conditions and U q 1/2 ( sl 2 ) trigonometric R−matrix. Indeed, in [14,15] it was discovered that the boundary analogue of U q 1/2 ( sl 2 ) is associated with the tridiagonal algebra, first introduced by Terwilliger [16].…”

“…However, a breakthrough recently emerged [14,15] from the study of the algebraic structure encoded in the reflection equation for certain (left purely non-diagonal and right diagonal) boundary conditions and U q 1/2 ( sl 2 ) trigonometric R−matrix. Indeed, in [14,15] it was discovered that the boundary analogue of U q 1/2 ( sl 2 ) is associated with the tridiagonal algebra, first introduced by Terwilliger [16]. This algebra is defined through a pair of tridiagonal relations; these are extensions of the q−Serre relations and can be viewed as "q−deformed" Dolan-Grady relations due to their close analogy with the relations discovered in [17].…”