Just as we know,
$w\_\infty$:=span {${z^\alpha }\partial \_z^\beta|\alpha,\beta\in\mathbb{Z}, \beta\geq0$ }.


But, what's the name of the following algebra,
span {$\{{z^{\alpha\_1}}{y^{\alpha\_2}}\partial \_z^{\beta\_1}\partial \_y^{\beta\_1}|\alpha\_i,\beta\_i\in\mathbb{Z}, \beta\geq0\}$ }?


Is it isomorphic to $w\_\infty \times w\_\infty$?


Can one provide some references about this algebra? Thanks very much!



What's the name of an algebra? is it isomorphic to $w_\infty \times w_\infty$?
If I'm not mistaken, this is the associative algebra of algebraic differential operators on the torus $\mathbb{G}\_{m,\mathbb{C}}^2$. It is the tensor product of two copies of $w\_\infty$, not the direct product. That is, you should replace $\times$ with $\otimes$.


