Representations and deformations of 3-Hom-ρ-Lie algebras

The aim of this paper is to introduce 3-Hom-[Formula: see text]-Lie algebra structures generalizing the algebras of 3-Hom-Lie algebra. Also, we investigate the representations and deformations theory of this type of Hom-Lie algebras. Moreover, we introduce the definition of extensions and abelian extensions of 3-Hom-[Formula: see text]-Lie algebras and show that associated to any abelian extension, there is a representation and a 2-cocycle.

TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

Genus fields of Kummer ℓn-cyclic extensions

We give a construction of the genus field for Kummer [Formula: see text]-cyclic extensions of rational congruence function fields, where [Formula: see text] is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer [Formula: see text]-cyclic extension. Finally, we study the extension [Formula: see text], for [Formula: see text], [Formula: see text] abelian extensions of [Formula: see text].