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 $\hslash ⟶0$ 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 $O({\hslash }^2)$, so that they are eigenfunctions of {\it two} immediately available commuting operators, arising from the $\hslash$-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 $|\lambda ⟩$, 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.

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Cited by in corpus (9)

• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Algorithms for representations of quiver Yangian algebras
• Macdonald polynomials for super-partitions
• Supersymmetric polynomials and algebro-combinatorial duality
• An Overview of Crystals and Double Quiver Yangians
• Vogel's universality and the classification problem for Jacobi identities
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• Super-Hamiltonians for super-Macdonald polynomials
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations