Simple representations of BPS algebras: the case of $$Y(\widehat{\mathfrak {gl}}_2)$$ Y ( gl ^ 2 )

Abstract

Abstract 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 $$Y(\widehat{\mathfrak {gl}}_r)$$ Y ( gl ^ r ) these representations are related to Uglov polynomials, whose families are also labeled by natural r. They arise in the limit $$\hbar {\longrightarrow } 0$$ ħ ⟶ 0 from Macdonald polynomials, and generalize the well-known Jack polynomials ( $$\beta $$ β -deformation of Schur functions), associated with $$r=1.$$ r = 1 . For $$r=2$$ r = 2 they approximate Macdonald polynomials with the accuracy $$O(\hbar ^2),$$ O ( ħ 2 ) , so that they are eigenfunctions of two immediately available commuting operators, arising from the $$\hbar $$ 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 $$Y(\widehat{\mathfrak {gl}}_2),$$ Y ( gl ^ 2 ) , where $$U^{(2)}$$ U ( 2 ) are associated with the states $$|\lambda {\rangle },$$ | λ ⟩ , parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $$p_{2n+1}$$ p 2 n + 1 can be expressed through mutually commuting operators from Yangian, however even time-variables $$p_{2n}$$ p 2 n are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.