On universal central extensions of Hom_Leibniz algebras

Universal central extensions in semi-abelian categories

“…Two counterexamples. Our first counterexample is borrowed from [9]. It shows that a category-here the category NAAlg K of non-associative algebras over a field K, which is a variety of Ω-groups-can be semi-abelian without having to satisfy condition (UCE).…”

“…Unlike for Leibniz or Lie algebras (or for associative ones), the bracket need not satisfy any additional conditions. We write NAAlg K for the category of non-associative algebras over K and remark that it coincides with the category of Hom-Leibniz algebras of which the twisting map is trivial (α " 0 in the notations of [22,9]) and with the category of magmas in the monoidal category pVect K , b, Kq. Note that Leib K , and hence also Lie K and Vect K , are subvarieties of NAAlg K .…”

“…In [9] the following situation is considered: morphisms g : C Ñ B and f : B Ñ A where as vector spaces, A, B and C are 2-, 3-and 4-dimensional with respective bases ta 1 , a 2 u, tb 1 , b 2 , b 3 u and tc 1 , c 2 , c 3 , c 4 u. Their brackets are generated by…”

Hom-Bol algebras

“…Some algebraic abstractions of this study are given in [20], [33], [34]. For more recent results regarding Hom-Lie algebras or Hom-Leibniz algebras, one may refer to [4], [5], [13]. For further information on other Hom-type algebras, one may refer to, e.g., [6], [8], [19], [21], [33], [34].…”