Affine Lie algebras on admissible half-integer levels
In 1988 \textit{V. Kac} and \textit{M. Wakimoto} [Proc. Natl. Acad. Sci. USA 85, No. 14, 4956--4960 (1988; Zbl 0652.17010)] introduced certain generalizations of highest weight integrable modules for affine Lie algebras, called admissible modules. Let \({\mathfrak g}\) be a finite-dimensional simple Lie algebra and \(l\) an admissible level. Then the highest weight \(\widehat{{\mathfrak g}}\)-module \(L_{{\mathfrak g}}(l,0)\) has a natural vertex operator algebra structure. Thus, it is of interest to describe the category of \(L_{{\mathfrak g}}(l,0)\)-modules. Since this category is too big for practical purposes, a more feasible goal is to describe \(L_{{\mathfrak g}}(l,0)\)-modules in the category \(\mathcal{O}\). More precisely, it is expected that at any admissibel level \(l\), this category is semisimple with finitely-many irreducibles. Apart from \({\mathfrak g}={\mathfrak {sl}}_2\), this is still an open problem (with several partial results obtained, some due to the author). In the present paper the author classfies irreducible \(L_{{\mathfrak g}}(n-s+\frac{1}{2},0)\)-modules, were \({\mathfrak g}\) is of type \(B_s\) and \(n\) is a positive integer. Even though the techniques used in the paper are fairly standard, some calculations turn out to be quite intricate. The paper is clearly written and almost self-contained.
scientific article; zbMATH DE number 5118814
scientific article