The famous Monster algebra \(V\) is a \(196884\)-dimensional commutative non-associative real
algebra with \(1\) that was used by Griess to provide in 1991 an explicit construction of the Monster
sporadic simple group \(M.\) Starting from the work of Frenkel, Lepowsky, and Meurman on the
Moonshine Conjecture, Borcherds introduced in 1992 the Monster vertex-operator
algebra \(V^\natural\), for which the Griess algebra \(V\) is the weight \(2\) component.
In 1996, Miyamoto observed that some idempotent elements, called Ising vectors, in the weight
\(2\) components of VOAs lead to involutory automorphisms. In 2007, his student Sakuma classified
all VOAs generated by two Ising vectors, showing that all of them are subalgebras in the Monster
VOA \(V^\natural\). Based on the work of Sakuma, Ivanov suggested axioms of a class of algebras
generalizing the Griess algebra \(V\).
In the talk, we will review these developments and then describe recent results on Ivanov algebras,
obtained in collaboration with Hall and Rehren.
