Title | **Gröbner bases in universal enveloping algebras of Leibniz algebras** |

Author(s) | Manuel A. Insua, Manuel Ladra |

Type | Article in Journal |

Abstract | We give a proof of the Poincaré–Birkhoff–Witt theorem for universal enveloping algebras of finite dimensional Leibniz algebras using Gröbner bases in a free associative algebra. We also construct Gröbner bases for two-sided ideals in universal enveloping algebras using the concept of Factor-Gröbner basis introduced by Nordbeck [Nordbeck, P., 2001. On the finiteness of Gröbner bases computation in quotients of the free algebra. Appl. Algebra Engrg. Comm. Comput. 11, 157–180]. Our approach differs from the one applied to PBW algebras or G-algebras since our algebras have zero divisors. We use this technique to obtain an algorithm to solve the ideal membership problem. |

Keywords | Leibniz algebra, Universal enveloping algebra, Poincaré–Birkhoff–Witt theorem, Gröbner bases |

ISSN | 0747-7171 |

URL |
http://www.sciencedirect.com/science/article/pii/S0747717108001430 |

Language | English |

Journal | Journal of Symbolic Computation |

Volume | 44 |

Number | 5 |

Pages | 517 - 526 |

Year | 2009 |

Note | Spanish National Conference on Computer Algebra |

Edition | 0 |

Translation |
No |

Refereed |
No |