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Original language | English |
---|---|

Pages (from-to) | 6535-6564 |

Number of pages | 38 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 9 |

Early online date | 20 Mar 2018 |

DOIs | |

Publication status | Published - Sep 2018 |

It is shown that over an arbitrary field there exists a nil algebra

R whose adjoint group is not an Engel group. This answers a question by

Amberg and Sysak from 1997. The case of an uncountable eld also answers

a recent question by Zelmanov.

In 2007, Rump introduced braces and radical chains A^{n+1} = A * A^n and

A^(n+1) = A^(n)* A of a brace A. We show that the adjoint group of a

finite right brace is a nilpotent group if and only if A^(n) = 0 for some n. We

also show that the adjoint group of a finite left brace A is a nilpotent

group if and only if A^n = 0 for some n. Moreover, if A is a finite brace

whose adjoint group is nilpotent then A is the direct sum of braces whose

cardinatities are powers of prime numbers. We also introduce a chain of ideals A^[n]

of a left brace A and then use it to investigate braces which satisfy A^n = 0

and A^(m) = 0 for some m; n.

We also describe connections between our results and braided groups and

the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equa-

tion. It is worth noticing that by a result by Gateva-Ivanova braces are in

one-to-one correspondence with braided groups with involutive braiding op-

erators.

R whose adjoint group is not an Engel group. This answers a question by

Amberg and Sysak from 1997. The case of an uncountable eld also answers

a recent question by Zelmanov.

In 2007, Rump introduced braces and radical chains A^{n+1} = A * A^n and

A^(n+1) = A^(n)* A of a brace A. We show that the adjoint group of a

finite right brace is a nilpotent group if and only if A^(n) = 0 for some n. We

also show that the adjoint group of a finite left brace A is a nilpotent

group if and only if A^n = 0 for some n. Moreover, if A is a finite brace

whose adjoint group is nilpotent then A is the direct sum of braces whose

cardinatities are powers of prime numbers. We also introduce a chain of ideals A^[n]

of a left brace A and then use it to investigate braces which satisfy A^n = 0

and A^(m) = 0 for some m; n.

We also describe connections between our results and braided groups and

the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equa-

tion. It is worth noticing that by a result by Gateva-Ivanova braces are in

one-to-one correspondence with braided groups with involutive braiding op-

erators.

- Engel group, nilpotent group, adjoint group of a ring, braces, nil rings, nil algebras, the Yang-Baxter equation.

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ID: 30855009