On boundary superalgebras

Abstract

We examine the symmetry breaking of super algebras due to the presence of appropriate integrable boundary conditions. We investigate the boundary breaking symmetry associated to both reflection algebras and twisted super Yangians. We extract the generators of the resulting boundary symmetry as well as we provide explicit expressions of the associated Casimir operators.

New reflection matrices for theUq(gl(m|n)) case

“…where apparently T ± ab ∈ End((C N ) ⊗N ). The boundary super algebra also turns out to be an exact symmetry of the double row transfer matrix, however the explicit proof as well as a detailed discussion will be presented elsewhere (see [40]).…”

“…It is worth mentioning that as in the case of the non super symmetric quantum algebras appropriate choice of boundary conditions leads to known quantum algebras as boundary symmetries. For instance, if we choose K ∝ I then one may show that the boundary algebra is the U q (gl(m|n)) (see [40] for details on the proof), which apparently is central to the super symmetric realization of the A-type Hecke algebra (U 0 in this case is trivial).…”

Temperley–LiebK-matrices

“…Although a hard task, direct computation has been used to solve (1.2). For instance, we mention the solutions with R matrix based in non-exceptional Lie algebras [22,26] and superalgebras [27,28]. The regular K-matrices for the exceptional U q [G 2 ] vertex model were obtained in [29].…”

On the \mathcal {U}_{q}[sl(2)] Temperley–Lieb reflection matrices

“…Although being a hard problem, the direct computation has been used to derive the solutions of the boundary Yang-Baxter equation (1.2) for given R. For instance, we mention the solutions with the R matrix based in non-exceptional Lie algebras [19,20] and superalgebras [21,22]. The regular K-matrices for the exceptional U q [G 2 ] vertex model were obtained in [23].…”