On tortkara triple systems

Abstract

The commutator [a, b] = ab − ba in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation T (a, b, c, d) which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying T (a, b, c, d) which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product [a, b, c] = [[a, b], c] in a free Zinbiel algebra and use computer algebra to construct a relation T T (a, b, c, d, e) which implies every relation satisfied by [a, b, c] in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by [a, b, c] which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity ≥ 9.

Triple Tortken Identities

“…To calculate triple Tortken identities in degree 5, we write a code on the software program Wolfram Mathematics and using it we demonstrate a list of polynomial identities of degree five by the triple Tortken product in every Novikov algebra. Triple identities of algebras were considered for many well-known classes of algebras such as Jordan, Lie [8] and Zinbiel algebras [2]. In [2], M. Bremner by using computer algebra studied special identities in terms of Tortkara triple product [𝑎𝑎, 𝑏𝑏, 𝑐𝑐] = [[𝑎𝑎, 𝑏𝑏], 𝑐𝑐] in a free Zinbiel algebra and discovered one identity in degree 5 and one identity in degree 7 which do not follow from the identities of lower degree.…”

On the speciality of Tortkara algebras

“…In general, we still do not know whether there exists a special identity. In [3], M. Bremner by computer algebraic methods has studied special identities in terms of Tortkara triple product [a, b, c] = [[a, b], c] in a free Zinbiel algebra and discovered one identity in degree 5 and one identity in degree 7 which do not follow from the triple identities of lower degree. We prove that there is no special identity in two variables.…”

The algebraic classification of nilpotent Tortkara algebras

“…Under the Koszul duality, the operad of Zinbiel algebras is dual to the operad of Leibniz algebras. Zinbiel algebras are also related to Tortkara algebras [15] and Tortkara triple systems [5]. More precisely, every Zinbiel algebra with the commutator multiplication gives a Tortkara algebra.…”