Simple Representations of BPS Algebras: the case of $Y(\widehat{\mathfrak{gl}}_2)$

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