On the constructions of holomorphic vertex operator algebras of central charge 24 (Q634622)

There is an ongoing interest in classification of vertex algebras of certain type. From this point of view, the most prominent algebras are of course holomorphic ones with \(c=24\), moonshine module \(V^\natural\) being the key example. A while ago \textit{C. Dong, G. Mason} and \textit{Y. Zhu} [Proc. Symp. Pure Math. 56, Part 2, 295--316 (1994; Zbl 0813.17019)] proved that the moonshine module is an example of what is now called a \textit{framed} VOA, meaning that it decomposes into a direct sum of the tensor product of minimal models of \(c=\frac{1}{2}\). In this paper several additional examples of framed holomorphic VOAs with \(c=24\) are constructed. It is then important to identify the graded component \(V_1\), which is known to be a semisimple or abelian Lie algebra. The author obtained an explicit decomposition in each case. These vertex algebras correspond to numbers \(7,10,18,19,26,33,35,40,48\) and 56 on the Schellekens' list [\textit{A. N. Schellekens}, Commun. Math. Phys. 153, 159--185 (1993; Zbl 0782.17014)].