# Algebras with skew-symmetric identity of degree 3

Article

DOI: 10.1007/s10958-009-9532-x

Dzhumadil’daev, A.S. J Math Sci (2009) 161: 11. doi:10.1007/s10958-009-9532-x

## Abstract

Algebras with one of the following identities are considered:
$$\begin{array}{*{20}{c}} {\left[ {\left[ {{t_1},\;{t_2}} \right],\;{t_3}} \right] + \left[ {\left[ {{t_2},\;{t_3}} \right],\;{t_1}} \right] + \left[ {\left[ {{t_3},\;{t_1}} \right],\;{t_2}} \right] = 0,} \\ {\left[ {{t_1},\;{t_2}} \right]{t_3} + \left[ {{t_2},\;{t_3}} \right]{t_1} + \left[ {{t_3},\;{t_1}} \right]{t_2} = 0,} \\ {\left\{ {\left[ {{t_1},\;{t_2}} \right],\;{t_3}} \right\} + \left\{ {\left[ {{t_2},\;{t_3}} \right],\;{t_1}} \right\} + \left\{ {\left[ {{t_3},\;{t_1}} \right],\;{t_2}} \right\} = 0,} \\ \end{array}$$
where [t1, t2] = t1t2− t2t1 and {t1, t2} = t1t2 + t2t1. We prove that any algebra with a skew-symmetric identity of degree 3 is isomorphic or anti-isomorphic to one of such algebras or can be obtained as their q-commutator algebras.