From affine Hecke algebras to boundary symmetries

Abstract

Motivated by earlier works we employ appropriate realizations of the affine Hecke algebra and we recover previously known non-diagonal solutions of the reflection equation for the $U_{q}(\hat{gl_n})$ case. The corresponding $N$ site spin chain with open boundary conditions is then constructed and boundary non-local charges associated to the non-diagonal solutions of the reflection equation are derived, as coproduct realizations of the reflection algebra. With the help of linear intertwining relations involving the aforementioned solutions of the reflection equation, the symmetry of the open spin chain with the corresponding boundary conditions is exhibited, being essentially a remnant of the $U_{q}(\hat{gl_{n}})$ algebra. More specifically, we show that representations of certain boundary non-local charges commute with the generators of the affine Hecke algebra and with the local Hamiltonian of the open spin chain for a particular choice of boundary conditions. Furthermore, we are able to show that the transfer matrix of the open spin chain commutes with a certain number of boundary non-local charges, depending on the choice of boundary conditions.Comment: 33 pages LATEX, few typos correcte

Integrable boundary conditions and modified Lax equations

“…From the algebraic perspective the two types of boundary conditions are associated with two distinct algebras, i.e. the reflection algebra [14] and the twisted Yangian respectively [31,32] (see also [25,29,30,33,34]). The classical versions of both algebras will be defined subsequently in the text (see section 3.2).…”

$K$-Матрицы Отражения, Связанные С $R$-Матрицами Темперли - Либа

“…Второй набор из m векторов размерности n позволяет построить вторую прямоугольную (n × m)-матрицу A максимального ранга m. Эта матрица A также определена с точностью до умножения справа на матрицу h ∈ GL(m). Легко видеть, что A t C = 0 в силу уравнения (12). Дополнительно отметим, что пересечение пространств V µ и V λ пусто, что эк-вивалентно требованию отсутствия векторов в пространстве V λ , удовлетворяющих уравнению (12), или требованию обратимости матрицы A t B.…”

On boundary superalgebras

“…We shall now make use of the generic relations, which clearly T satisfies due to the particular choice of boundary conditions (see also [7]):…”

“…We have not seen, as far as we know, such an explicit and elegant proof elsewhere not even in the non super symmetric case. In [7] the proof is explicit, but rather tedious, whereas in [2] one has to realize that the emanating non local charges form the U q (gl(N )), and this is not quite obvious. Note that T ± ab are quadratic combinations of the algebra generators, and in fact the corresponding super-trace provides the associated Casimir, which again gives a hint about the associated exact symmetry.…”

“…. , N − 1, and the exchange relations of the U q (gl(m|n)) algebra are given below: l + ij ,l − ij , i < j and l − ij ,l + ij , i > j are also non zero, and are expressed as combinations of the U q (gl(m|n)) generators, but are omitted here for brevity, see e.g expressions in [35,7] for the U q (gl(n)) case.…”