Gluing affine Yangians with bi-fundamentals
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Abstract
A bstract The affine Yangian of $$ {\mathfrak{gl}}_1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>gl</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> is isomorphic to the universal enveloping algebra of $$ {\mathcal{W}}_{1+\infty } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mo>∞</mml:mo> </mml:mrow> </mml:msub> </mml:math> and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 supersymmetric $$ {\mathcal{W}}_{\infty } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mo>∞</mml:mo> </mml:msub> </mml:math> algebra was constructed by “gluing” two affine Yangians of $$ {\mathfrak{gl}}_1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>gl</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> using operators that transform as (□, $$ \overline{\square} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mo>□</mml:mo> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ) and ( $$ \overline{\square} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mo>□</mml:mo> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , □) w.r.t. the two affine Yangians. In this paper we realize a similar (but non-isomorphic) two-parameter gluing construction where the gluing operators transform as (□, □) and ( $$ \overline{\square} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mo>□</mml:mo> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , $$ \overline{\square} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mo>□</mml:mo> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> ) w.r.t. the two affine Yangians. The corresponding representation space consists of pairs of plane partitions connected by a common leg whose cross-section takes the shape of Young diagrams, offering a more transparent geometric picture than the previous construction.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
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| Bibliometrics | 0.000 | 0.000 |
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| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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