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) Y ( gl ^ r ) these representations are related to Uglov polynomials, whose families are also labeled by natural r . They arise in the limit 0 ħ ⟶ 0 from Macdonald polynomials, and generalize the well-known Jack polynomials ( β β -deformation of Schur functions), associated with $$r=1.$$ r = 1 . For $$r=2$$ r = 2 they approximate Macdonald polynomials with the accuracy O( ²), O ( ħ 2 ) , so that they are eigenfunctions of two immediately available commuting operators, arising from the ħ -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), Y ( gl ^ 2 ) , where U⁽²⁾ U ( 2 ) are associated with the states |λ , | λ ⟩ , parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables p₂ₙ₊₁ p 2 n + 1 can be expressed through mutually commuting operators from Yangian, however even time-variables p₂ₙ p 2 n are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.