# On tortkara triple systems

## TL;DR

This paper investigates the algebraic structures called tortkara triple systems derived from Zinbiel algebras, using computer algebra to find identities and relations in various arities, revealing their operadic properties and open questions.

## Contribution

It introduces new identities for tortkara triple systems, showing their quadratic ternary operad structure and providing computational methods to explore higher arity relations.

## Key findings

- Constructed a relation in arity 5 that implies all relations in arity 5.
- Identified a relation in arity 7 not implied by lower arity relations.
- Demonstrated that tortkara triple systems are governed by a quadratic ternary operad with a Koszul dual.

## 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 $TT(a,b,c,d,e)$ 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 $\ge 9$.