Article

Journal of Mathematical Sciences

, 161:11

First online:

Algebras with skew-symmetric identity of degree 3

  • A. S. Dzhumadil’daevAffiliated withKazakh-British Technical University Email author 

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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 [t 1 , t 2] = t 1 t 2 − t 2 t 1 and {t 1, t 2} = t 1 t 2 + t 2 t 1 . 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.