Classification of division \({\mathbb Z}^n\)-graded alternative algebras.
The octonion torus appears as a coordinate algebra of extended affine Lie algebras of type \(A_2\) and \(F_4\). The author classifies a generalized version of the octonion torus, called division \(\mathbb Z^n\)-graded alternative algebras. It turns out that they are strongly prime and so by a result due to Slater it is either associative or an octonion ring, i. e., its central closure is an octonion algebra. The author identifies four specific division \(\mathbb Z^n\)-graded octonion rings, and he obtains that a division \(\mathbb Z^n\)-graded octonion ring is isomorphic to exactly one of these. Using this result, the author completes the classification of division \((A_2,\mathbb Z^n)\)-graded Lie algebras, up to central extensions, which are a generalization of the cores of extended affine Lie algebras of type \(A_2\).
division \(\mathbb Z^n\)-graded alternative algebra
scientific article; zbMATH DE number 1868019