Mathematicians unlock hidden structures in abstract algebra with practical applications

<p>The Freudenthal–Tits magic square  for  of semi-simple Lie algebras can be extended by including the sextonions . A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the intermediate exceptional series, with the largest one as the intermediate Lie algebra  constructed by Landsberg–Manivel. We study various aspects of the intermediate vertex operator (super)algebras associated with the intermediate exceptional series, including rationality, coset constructions, irreducible modules, (super)characters and modular linear differential equations. For all  belonging to the intermediate exceptional series, the intermediate VOA  has characters of irreducible modules coinciding with those of the simple rational -cofinite W-algebra  studied by Kawasetsu, with  belonging to the Cvitanović–Deligne exceptional series. We propose some new intermediate VOA  with integer level k and investigate their properties. For example, for the intermediate Lie algebra  between  and  in the subexceptional series and also in Vogel’s projective plane, we find that the intermediate VOA  has a simple current extension to a SVOA with four irreducible Neveu–Schwarz modules, and the supercharacters can be realized by a fermionic Hecke operator on the  Virasoro minimal model . We also provide some (super) coset constructions such as  and . In the end, we find that the theta blocks associated with the intermediate exceptional series produce some new holomorphic Jacobi forms of critical weight and lattice index.</p>