3-dimensional associative noncommutative algebras (Q1841826)

It is proved that the vector product, over any field \(k\), cannot be represented as a commutator \(xy-yx\) in a \(3\)-dimensional associative algebra. Here the vector product on \(k^3\) is defined by \[ (x_1,x_2,x_3)\times (y_1,y_2,y_3) =(x_2y_3-x_3y_2,x_3y_1-x_1y_3,x_1y_2-x_2y_1). \] It satisfies skew-symmetry and the Jacobi identity. Hence the \(k\)-vector space \(k^3\) is also equipped with a Lie bracket. The Lie algebra obtained is isomorphic to \({\mathfrak so}_3(k)\). In this context the author's result shows that there are no compatible associative structures on \({\mathfrak so}_3(k)\). If the characteristic of \(k\) is zero then there is not even a compatible left-symmetric structure on \(\mathfrak{so}_3(k)\). This is not true in general for fields of finite characteristic.