Boundary Lax pairs from non-ultra-local Poisson algebras

Abstract

We consider non-ultra local linear Poisson algebras on a continuous line . Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or "boundary" extensions. They are parametrized by a "boundary" scalar matrix and depend in addition on the choice of an antiautomorphism. The new algebras are the classical-linear counterparts of known quadratic quantum boundary algebras. For any choice of parameters the non-ultra local contribution of the original Poisson algebra disappears. We also systematically construct the associated classical Lax pair. The corresponding discrete linear algebra is constructed, and the classical boundary discrete and continuum PCM models are examined as physical examples.

Generalized Landau–Lifshitz models on the interval

“…It is shown that the presence of special boundary terms suitably breaks the symmetry of the models under consideration, as also happens in discrete integrable models (see e.g. [5,8,27]).…”

“…i,j P 0i P 0j → dx dy P 0 (x)P 0 (y). (7.5) The modified monodromy matrix as well as the B, P matrices may be expressed as The entries T kl of the matrix are the non-local charges, which are realizations of the classical reflection algebra (3.3), (see also [27]). In particular, the continuum non-local charges emerging from (7.4) via (7.5) become: Consider now the continuum expression for T with T given in (5.10), and take into account that for the L-L model…”

Poisson color algebras of arbitrary degree

“…Now we have to note that another class of Poisson-type algebras similar to the one of graded Poisson algebras of degree z 0 in Definition 1 but replacing the group Z by Z 2 has been considered in the literature. This kind of algebras are known as even and odd Poisson superalgebras, depending on taking degree0 or degree1, being of interest in studying, for instance, two-dimensional supergravity and three-dimensional systems ( [2,11,14,23]). However, as we know, there is not a category in the literature which allows us to combine a graded bracket of degree g 0 ∈ G and a graded commutative associative product via a graded Leibniz identity when the group G is an arbitrary abelian group.…”

“…As another example, we note that it is possible to recover Hamiltonian mechanics from the coordinate space of the theory by making use of graded Poisson algebras ( [20]). We can enumerate many more applications (see [2,14,15,18,23]), but we refer to [10] to a good review on this matter. Definition 1.…”

Strongly graded Poisson-Hilbert spaces

“…We also note that the interest on gradings on different classes of algebras has been remarkable in the last years, specially motivated by their applications in physics and geometry, see for instance [5,9,12,13,14,16,21]. Now recall that a Poisson algebra is an associative algebra endowed with a Lie product, denoted by {•, •}, in such a way that the Lie and the associative products are compatible via the Leibniz identity {x, yz} = {x, y}z + y{x, z} for any x, y, z ∈ P (see [1,4,17,18]). Also recall that the context of strong graduation has been extensively considered in the literature (see for instance the text book [20] or the recent reference [15,22,23]).…”