SIGMA 22 (2026), 052, 16 pages      arXiv:2512.02718      https://doi.org/10.3842/SIGMA.2026.052
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

Irreducibility of Certain $\widehat{\mathfrak{sl}}_2$-Modules of Wakimoto Type

Dražen Adamović and Veronika Pedić Tomić
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, Zagreb, Croatia

Received December 03, 2025, in final form May 11, 2026; Published online May 21, 2026

Abstract
We investigate the irreducible smooth $\widehat{\mathfrak{sl}}_{2}$-modules recently constructed in [Adv. Math. 481 (2025), 110559, 34 pages, arXiv:2404.03855], and demonstrate that these modules admit a Wakimoto-type realization at both critical and non-critical levels. In the critical level case, we identify simple quotients of these modules with the Wakimoto modules whose irreducibility was already established by Adamović. We also generalize some Wakimoto modules constructed in [Adv. Math. 289 (2016), 438-479, arXiv:1409.5354] and identify them as generalized Whittaker modules.

Key words: vertex algebra; Whittaker modules; Wakimoto modules; smooth modules.

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