Classification of nilpotent associative algebras of small dimension

Abstract

Abstract. We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.

The Algebraic and Geometric Classification of Nilpotent Bicommutative Algebras

“…Using this method, all the non-Lie central extensions of all 4-dimensional Malcev algebras were described afterwards [29], and also all the non-associative central extensions of 3-dimensional Jordan algebras [28], all the anticommutative central extensions of the 3-dimensional anticommutative algebras [8], and all the central extensions of the 2-dimensional algebras [10]. Moreover, the method is especially indicated for the classification of nilpotent algebras (see, for example, [26]), and it was used to describe all the 4-dimensional nilpotent associative algebras [15], all the 4-dimensional nilpotent Novikov algebras [33], all the 5-dimensional nilpotent Jordan algebras [27], all the 5-dimensional nilpotent restricted Lie algebras [13], all the 6-dimensional nilpotent Lie algebras [12,14], all the 6-dimensional nilpotent Malcev algebras [30] and some others.…”

The algebraic classification of nilpotent Tortkara algebras

“…After that, the method introduced by Skjelbred and Sund was used to describe all non-Lie central extensions of all 4-dimensional Malcev algebras [21], all non-associative central extensions of 3-dimensional Jordan algebras [20], all anticommutative central extensions of 3-dimensional anticommutative algebras [6], all central extensions of 2-dimensional algebras [7]. The method of central extensions was used to describe all 4-dimensional nilpotent associative algebras [12], all 4-dimensional nilpotent bicommutative algebras [26], all 4-dimensional nilpotent Novikov algebras [24], all 5-dimensional nilpotent Jordan algebras [19], all 5-dimensional nilpotent restricted Lie algebras [11], all 6-dimensional nilpotent Lie algebras [10,13], all 6-dimensional nilpotent Malcev algebras [22], all 6-dimensional nilpotent binary Lie algebras [3], all 6-dimensional nilpotent anticommutative CD-algebras [3] and some other.…”

Some Classes of Nilpotent Associative Algebras

“…Poonen [16] analyzed nilpotent commutative associative algebras of dimension less and equal 5 over any algebraic closed field and Eick and Moede [6,7] initiated on an arbitrary field a coclass theory for nilpotent associative algebras, giving a different view on their classification since they use the coclass as primary invariant, describing algorithms that allow to investigate the graph associated with the nilpotent associative algebras of a certain coclass for a finite field. Recently, De Graaf classified nilpotent associative algebras of dimension less and equal than 4 by using method of central extensions [5].…”