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

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 $$Y(\widehat{\mathfrak {gl}}_r)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mover> <mml:mi>gl</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mi>r</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> these representations are related to Uglov polynomials, whose families are also labeled by natural r . They arise in the limit $$\hbar {\longrightarrow } 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ħ</mml:mi> <mml:mo>⟶</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> from Macdonald polynomials, and generalize the well-known Jack polynomials ( $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -deformation of Schur functions), associated with $$r=1.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> For $$r=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> they approximate Macdonald polynomials with the accuracy $$O(\hbar ^2),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ħ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> so that they are eigenfunctions of two immediately available commuting operators, arising from the $$\hbar $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ħ</mml:mi> </mml:math> -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 $$Y(\widehat{\mathfrak {gl}}_2),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mover> <mml:mi>gl</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where $$U^{(2)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> are associated with the states $$|\lambda {\rangle },$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>⟩</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $$p_{2n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> can be expressed through mutually commuting operators from Yangian, however even time-variables $$p_{2n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> are inexpressible. Implications to higher r become now straightforward, yet we describe them only in a sketchy way.

Topics & Concepts