## Abstract

Abstract: This paper investigates whether a root lattice can be similar to the lattice θ of all integer elements of a number field K endowed with the inner product (x, y):=Trace_{K /Q}(x.θ(y)), where θ is an involution of the field K. For each of the following three properties (1), (2), (3), a classification of all the pairs K, θ with this property is obtained: (1) θ is a root lattice; (2) θ is similar to an even root lattice; (3) θ is similar to the lattice ℤ[_{K:Q}]. The necessary conditions for similarity of θ to a root lattice of other types are also obtained. It is proved that θ cannot be similar to a positive definite even unimodular lattice of rank ≤48, in particular, to the Leech lattice.

Original language | English (US) |
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Pages (from-to) | 221-223 |

Number of pages | 3 |

Journal | Doklady Mathematics |

Volume | 101 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2020 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)