A new class of simple smooth modules over the affine algebra \(A_1^{(1)}\) (Q6597517)

The article presents a novel family of modules for the affine special linear Lie algebra of rank~1, with the stated properties. (A simple module is one with no nontrivial submodule, and a smooth module is such that, with respect to a $\mathbb{Z}$ grading, all homogeneous elements above some threshold $N$ act as 0 on the module.) The authors nicely situate their work in the context of the much-studied classes of modules for a semisimple Lie algebra: highest weight modules and Whittaker modules, both of which are smooth.\N\NAfter recalling some background about affine Kac-Moody algebras and Whittaker and smooth modules for such Lie algebras, the authors define what amount to the fundamental building blocks of their subsequent argument: one is a Lie subalgebra of $\widehat{\mathfrak{sl}(2)}$, the universal central extension of $\mathfrak{sl}(2)$, labeled $\mathcal{P}_N$. From $\mathcal{P}_N$ and a one-dimensional $(\mathcal{P}_N \oplus \mathbb{C}c)$-module, a certain induced module $W_{\phi}$ (induced from the universal enveloping algebra of $(\mathcal{P}_N \oplus \mathbb{C}c)$ up to the universal enveloping algebra of $\widehat{\mathfrak{sl}(2)}$) is the basic object in which the authors are interested.\N\NThe first main result, Theorem~3.2, gives the conditions in which $W_{\phi}$ is a simple $\widehat{\mathfrak{sl}(2)}$-module. The ``only if'' portion of the statement is proved quickly in Lemma~3.3, while the ``if'' portion requires a sequence of rather technical lemmas, culminating in Lemma~3.12. The proof uses weight and length arguments for large sums/products of elements in the universal enveloping algebra of $\widehat{\mathfrak{sl}(2)}$. Theorem~3.13 provides conditions in which $W_{\phi}$ are isomorphic.\N\NSection 4 adds to $\widehat{\mathfrak{sl}(2)}$ the degree derivation $d$, which the authors notate $\tilde{\mathfrak{sl}}(2)$. With the Casimir operator to give a filtration of $W_{\phi}[d]$, they use the results of Section~3 to produce simple modules for $\tilde{\mathfrak{sl}}(2)$, Whittaker vectors of $W_{\phi}[d]$, and isomorphism conditions for simple $\tilde{\mathfrak{sl}}(2)$-subquotients of $W_{\phi}[d]$.\N\NIn addition to the article's cited references, a few relevant related works include the following list:\N[\textit{H.~Chen} et al., Adv. Math. 454, Article ID 109874, 60~p. (2024; Zbl 07900777)],\N[\textit{V.~Futorny} et al., ``Smooth representations of affine Kac-Moody algebras'', Preprint, \url{arXiv:2404.03855}],\N[\textit{L.~Ge} and \textit{Z.~Li}, J. Algebra 644, 23--63 (2024; Zbl 1547.17046)],\N[\textit{H.~Matumoto}, Duke Math. J. 60, No.~1, 59--113 (1990; Zbl 0716.17007)].