Self-dual lattices of type \(A\)
The author generalizes \textit{I. Frenkel}, \textit{J. Lepowsky} and \textit{A. Meurman}'s construction [Proc. Natl. Acad. Sci. USA 81, 3256-3260 (1984; Zbl 0543.20016) and Vertex Operator Algebras and the Monster, Academic Press (1988; Zbl 0674.17001)] of the moonshine module for the Monster finite simple group. A key step in those authors' work was the ``triality'' construction, which involves certain modules for a vertex operator algebra corresponding to an integral lattice formed by gluing together finitely many copies of the root lattice of type \(A_1\). This author's investigation of the existence of other vertex operator algebras with finite automorphism groups corresponds to the group-theoretic question of the existence of additional finite groups with moonshine representations analogous to the Monster's. In related papers [Commun. Algebra 22, 1281-1303 (1994; Zbl 0820.17028) and J. Pure Appl. Algebra 99, 53-111 (1995; Zbl 0838.17032)] the author has investigated central extensions of these lattices and has constructed an analogue of ``vertex operator triality'' for each self-orthogonal ternary code (that is, code over \(\mathbb{Z}_3)\) that contains \((1, \dots, 1)\). The author generalizes Frenkel, Lepowsky and Meurman's ``\(\mathbb{Z}_2\)-orbit fold technique'' to \(\mathbb{Z}_n\) for \(n \geq 2\) to produce vertex operator algebras that correspond to lattices related to finitely many copies of a root lattice of type \(A_{n - 1}\). Specifically, the paper develops two gluing techniques for constructing self-dual lattices. Given a rational lattice \(L\) (a free \(\mathbb{Z}\)-module of finite rank with a \(\mathbb{Q}\)-valued symmetric \(\mathbb{Z}\)-bilinear form), its integral dual is \(L^0 = \{y \in L_\mathbb{Q} \mid \langle y,x \rangle \in \mathbb{Z}\) for all \(x \in L\}\). The lattice is self-dual if \(L = L^0\). A linear code of length \(k\) over \(\mathbb{Z}_n\) is a \(\mathbb{Z}_n\)-submodule \({\mathcal C}\) of \(\mathbb{Z}^k_n\). For a symmetric bilinear form \(f\) on \(\mathbb{Z}^k_n\) the dual code of \({\mathcal C}\) is \({\mathcal C}^\perp_f = \{\alpha \in \mathbb{Z}^k_n \mid f (\alpha, \beta) = 0\) for all \(\beta \in {\mathcal C}\}\). A code is self-orthogonal (respectively, self-dual) relative to \(f\) if \({\mathcal C} \subset {\mathcal C}^\perp_f \) (respectively, \({\mathcal C} = {\mathcal C}^\perp_f)\). (If \(f\) is associated with the identity matrix, then ``relative to \(f\)'' is omitted.) The approach of the paper grows from constructions of self-dual lattices by \textit{J. H. Conway} and \textit{N. J. A. Sloane} [Proc. R. Soc. Lond., Ser. A 381, 275-283 (1982; Zbl 0496.10021) and J. Number Theory 15, 83-94 (1982; Zbl 0496.10023)] and earlier work of \textit{N. J. A. Sloane} [Proc. Symp. Pure Math. 34, 273-308 (1979; Zbl 0407.94009)]. The author refines the gluing procedure of \textit{J. H. Conway} and \textit{N. J. A. Sloane} in Chapter 4 of their monograph ``Sphere packings, lattices and groups'', Springer (1993; Zbl 0785.11036) and (1988; Zbl 0634.52002). These techniques produce two families of root lattices of type \(A\) that are based on the ring structure of a root lattice of type \(A\) induced by the Coxeter element.
scientific article; zbMATH DE number 807539
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