Naturally graded Zinbiel algebras with nilindexn−3

Degenerations of Zinbiel and nilpotent Leibniz algebras

“…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".…”

The algebraic classification of nilpotent Tortkara algebras

“…Theorem 8. Let Tort 3 be the variety of all 3-dimensional Tortkara algebra over C and T be a nonzero algebra from Tort 3 , then (1) T is isomorphic to one of the following algebras: • g 1 : e 2 e 3 = e 1 • g 2 : e 1 e 3 = e 1 , e 2 e 3 = e 2 • g α 3 : e 1 e 3 = e 1 + e 2 , e 2 e 3 = αe 2 • A 2 : e 1 e 2 = e 1 , e 2 e 3 = e 2 • A 0 1 : e 1 e 2 = e 3 , e 1 e 3 = e 1 + e 3 , e 2 e 3 = αe 2 (2) The variety of 3-dimensional Tortkara algebras has one irreducible component defined by the rigid algebra A 0 1 .…”

“…These currents span an algebra which is closely related to so called affine Kac-Moody algebras, which are universal central extensions of loop algebras. Central extensions are needed in physics, 1 The work was supported by RSF 18-71-10007. The authors thank Pilar Páez-Guillán for some constructive comments.…”

“…The authors thank Pilar Páez-Guillán for some constructive comments. 2 Corresponding Author: kaygorodov.ivan@gmail.com 1 because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra. Kac-Moody algebras have been conjectured to be a symmetry groups of a unified superstring theory.…”

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

“…Later on, naturally graded Leibniz algebras of nilindex n − 2 that admit the following characteristic sequences (n − 3, 3), (n − 3, 2, 1), (n − 3, 1, 1, 1) were investigated in a series of papers [6,7,11], respectively. Description of naturally graded Zinbiel algebras with these properties was given in [1] and [2].…”