On tortkara triple systems
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arXiv
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2017
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| author | Bremner, Murray |
| author_facet | Bremner, Murray |
| contents | 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 $\ge 9$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1706_03748 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On tortkara triple systems Bremner, Murray Rings and Algebras Quantum Algebra Representation Theory Primary 17A40. Secondary 15-04, 15A72, 17-04, 17A30, 17A32, 17A50, 18D50, 20C30, 20-04, 68W30 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 $\ge 9$. |
| title | On tortkara triple systems |
| topic | Rings and Algebras Quantum Algebra Representation Theory Primary 17A40. Secondary 15-04, 15A72, 17-04, 17A30, 17A32, 17A50, 18D50, 20C30, 20-04, 68W30 |
| url | https://arxiv.org/abs/1706.03748 |