Abelsche

Abelsche is an adjective used in German mathematical terminology to denote properties, structures, or results associated with the mathematician Niels Henrik Abel or, more commonly, with the concept of abelian (commutative) structures. The form appears in several standard terms, most notably Abelsche Gruppe, the German name for an Abelian group, which is a group (G, *) in which a*b = b*a for all a, b in G. The commutativity condition gives rise to a rich theory and simpler algebraic behavior.

In algebraic geometry and related fields, Abelsche Varietät (Abelian variety) refers to a complete algebraic variety equipped with a group law that is compatible with its geometric structure. One-dimensional examples are elliptic curves, which are fundamental objects in number theory and algebraic geometry.

Historically, the term reflects the work and legacy associated with Abel and his contributions to the study of algebraic equations and functions; many theorems bear his name, and the adjective Abelsche is used to indicate a connection to that lineage or to the broader notion of commutativity in algebra.

Usage notes: The English equivalent is typically Abelian, as in Abelian group or Abelian variety. In German mathematical literature, Abelsche Gruppe and Abelsche Varietät are standard terms, with capitalization as appropriate for the object and the rest of the phrase following ordinary adjective inflection.

See also: Abel, Abelian group, Abelian variety, Abelianization, Abel’s theorem.