Classifications of some classes of Zinbiel algebras

Abstract

In this work nul-filiform and filiform Zinbiel algebras are described up to isomorphism. Moreover, the classification of complex Zinbiel algebras is extended from dimensions ≤ 3 up to the dimension 4.

Degenerations of Zinbiel and nilpotent Leibniz algebras

“…The classification of nilpotent four-dimensional Leibniz algebras is given in [4]. The classification of four-dimensional Zinbiel algebras is given in [2]. We denote by N the variety of three-step nilpotent associative four-dimensional (Leibniz-Zinbiel) algebras, by Z the variety of Zinbiel four-dimensional algebras, and by L the variety of nilpotent Leibniz four-dimensional algebras.…”

“…We describe all four-dimensional Zinbiel and nilpotent Leibniz algebras in Table 1 below. We use the classification results of [2,4,17,27] for this. Only the algebras N 2 1 and L 1 didn't appear in these classifications.…”

“…We use the letters N, Z, and L in our notation to indicate algebras from N, Z \ N, and L \ N respectively. The second, the third, and the fourth columns of Table 1 contain the names from [2,4,17,27], the multiplication tables, and the dimensions of algebras of derivations respectively for the algebras under consideration. We omit in multiplication tables zero products of basic elements.…”

“…Under the Koszul duality the operad of Zinbiel algebras is dual to the operad of Leibniz algebras. Zinbiel algebras were studied in [1,2,13,16,17]. 1 The work was supported by RFBR 16-31-50017, 14-01-00014, FAPESP 14/19521-3, 14/24519-8 and by R & D 6.38.191.2014 of Saint-Petersburg State University, "Structure theory, classification, geometry, K-theory and arithmetics of algebraic groups and related structures".…”

Classification of Naturally Graded Zinbiel Algebras with Characteristic Sequence Equal to (n–p, p)

“…For instance, n-dimensional Leibniz algebras of nilindices n + 1 and n (which is equivalent to admit characteristic sequences equal to (n) and (n − 1, 1), respectively) were described in papers [5] and [14]. Similar description for Zinbiel algebras were obtained in the paper [4].…”