On the speciality of Tortkara algebras

Abstract

A criterion for elements of free Zinbiel algebras to be Lie or Jordan is established. This criterion is used in studying speciality problems of Tortkara algebras. We construct a base of free special Tortkara algebras. Furthermore, we prove analogues of classical Cohn's and Shirshov's theorems for Tortkara algebras.2000 Mathematics Subject Classification. 17A30, 17A50.

Time-Warping Invariants of Multidimensional Time Series

“…One can show, however, that area satisfies a fourth-order relation, known as tortkara, introduced by Dzhumadil'daev in the 2007 paper [14]. In [15] the image of iterated applications of the area map is characterised. (Compare also [45,Theorem 28]).…”

“…[15, Theorem 2.1]) The space D is spanned by the setA ∪ { u([a][b] − [b][a]) : a, b ∈ A, u ∈ T (A) }.From Lemma 2.7 and Lemma 2.8 we deduce the following morphism property of Hoffman's isomorphism with respect to area and area.Theorem 5.9 H : D → D is a tortkara morphism, i.e., for ϕ, ψ ∈ D H (area(ϕ, ψ)) = area( H (ϕ), H (ψ)).…”

Gelfand–Dorfman algebras, derived identities, and the Manin product of operads

“…In [7], a linear basis of the free Novikov algebra generated by an ordered set was described in terms of partitions and Young diagrams. To prove the statement, we use this basis that consists of nonassociative monomials constructed by a Young diagram with a proper filling the cells by the generators, see [8,Section 4] for details.…”

Symmetric Zinbiel superalgebras

“…Zinbiel algebras also appeared in a study of rack cohomology [12], number theory [10] and in a construction of a Cartesian differential category [24]. In recent years, there has been a strong interest in the study of Zinbiel algebras in the algebraic and the operad context [4,8,15,23,16,17,18,21,26,30,32,33,34].…”