The affine Yangian of gl 1 $$ {\mathfrak{gl}}_1 $$ is isomorphic to the universal enveloping algebra of W 1 + ∞ $$ {\mathcal{W}}_{1+\infty } $$ and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of N $$ \mathcal{N} $$ = 2 supersymmetric W ∞ $$ {\mathcal{W}}_{\infty } $$ algebra was constructed by “gluing” two affine Yangians of gl 1 $$ {\mathfrak{gl}}_1 $$ using operators that transform as (□, □ ¯ $$ \overline{\square} $$ ) and ( □ ¯ $$ \overline{\square} $$ , □) 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 ( □ ¯ $$ \overline{\square} $$ , □ ¯ $$ \overline{\square} $$ ) 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.

中文翻译：

---

用双基本面粘合仿射阳线

gl 1 $$ {\mathfrak{gl}}_1 $$ 的仿射杨安与 W 1 + ∞ $$ {\mathcal{W}}_{1+\infty } $$ 的泛包络代数同构，并且可以作为构建新顶点算子代数的构建块。在[1]中，N $$ \mathcal{N} $$ = 2 超对称 W ∞ $$ {\mathcal{W}}_{\infty } $$ 代数的二参数族推广是通过“粘合”构造的gl 1 $$ {\mathfrak{gl}}_1 $$ 的两个仿射杨氏使用转换为 (□, □¯ $$ \overline{\square} $$ ) 和 ( □¯ $$ \overline{\square } $$ , □) 写了两个仿射扬安人。在本文中，我们实现了一个类似（但非同构）的双参数粘合构造，其中粘合运算符转换为 (□, □) 和 ( □¯ $$ \overline{\square} $$ , □¯ $$ \overline {\square} $$ ) 是两个仿射杨安派。