On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0

Area of Science:

• Algebraic Geometry
• Group Theory
• Commutative Algebra

Background:

• Connected commutative algebraic groups over fields of characteristic 0 are fundamental objects in algebraic geometry.
• Morphisms between these groups can be studied both as algebraic variety maps and as group homomorphisms.

Purpose of the Study:

• To construct a natural retraction from the set of variety morphisms (preserving the identity) to the set of group homomorphisms.
• To investigate the implications of this retraction for the isomorphism of algebraic groups and varieties.
• To characterize variety automorphisms of specific algebraic groups.

Main Methods:

• Construction of a retraction map from Mor0(G,H) to Hom(G,H).
• Analysis of the properties of this retraction, including its compatibility with composition and addition.
• Explicit description of morphisms and isomorphisms for algebraic groups without non-trivial unipotent factors.

Main Results:

• A natural retraction is established between variety morphisms and group homomorphisms for connected commutative algebraic groups.
• If two such groups are isomorphic as varieties, they are also isomorphic as algebraic groups.
• An explicit description of morphisms and isomorphisms is provided for a class of algebraic groups.

Conclusions:

• The constructed retraction provides a unified framework for studying morphisms between algebraic groups.
• The results clarify the relationship between variety isomorphism and algebraic group isomorphism.
• The characterization of automorphisms offers insights into the structure of specific algebraic groups.