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 (β-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 |λ, parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables p₂ₙ₊₁ can be expressed through mutually commuting operators from Yangian, however even time-variables p₂ₙ are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.