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