On the triality of the Hamming code vertex operator algebra
\textit{M. Miyamoto} introduced the notion of code vertex operator algebra [J. Algebra 181, 207-222 (1996; Zbl 0857.17026)]. The code vertex operator algebra \(V_D\) associated to an even binary linear code \(D\) is a sum of tensor products of the Ising model, the irreducible unitary highest weight representations of the Virasoro algebra of central charge \(1/2\), where the number of the factors is the length of the code \(D\), and the summands are in one-to-one correspondence with \(D\) by forming the tensor product of \(L(1/2,0)\)'s and \(L(1/2, 1/2)\)'s according codewords. Moreover, Miyamoto proved that there is an extra involution automorphism for each conformal vector in such an algebra. Denote by \(H_8\) the \([8,4,4]\) extended binary Hamming code and by \(V(D_4)\) the vertex operator algebra associated with the root lattice of the simple Lie algebra of type \(D_4\). The author of this paper constructs an explicit embedding of \(V_{H_8}\) into \(V(D_4)\) so that the extra involutions are obtained by restricting the automorphism of \(V(D_4)\) that corresponds to the classical triality of \(D_4\).
scientific article; zbMATH DE number 1526419
scientific article