Simple Representations of BPS Algebras: the case of Y (gl₂)

BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians -- the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for Y (glᵣ) these representations are related to Uglov polynomials, whose families are also labeled by natural r. They arise in the limit 0 from Macdonald polynomials, and generalize the well-known Jack polynomials (-deformation of Schur functions), associated with r=1. For r=2 they approximate Macdonald polynomials with the accuracy O (²), so that they are eigenfunctions of two immediately available commuting operators, arising from the -expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, -- what provides a technically simple way to build an explicit representation of Yangian Y (gl₂), where U^ (2) are associated with the states |, parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables p₂₍+₁ can be expressed through mutually commuting operators from Yangian, however even time-variables p₂₍ are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.