2-Dimensional Algebras Application to Jordan,G-Associative and Hom-Associative Algebras

Abstract

We classify, up to isomorphism, the 2-dimensional algebras over a field . We focuse also on the case of characteristic 2, identifying the matrices of GL(2,  2 ) with the elements of the symmetric group ∑ 3 . The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.