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In Signal Flow Graph (SFG) algebra, the value a node represents is determined by the sum of all signals entering that node. This summed value is then transmitted through every branch leaving the node, making the SFG a powerful tool for visualizing and analyzing control systems.
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On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0.

Gabriel A Dill1

  • 1Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167 Germany.

Transformation Groups
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PubMed
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This study introduces a natural retraction for morphisms between connected commutative algebraic groups, simplifying the understanding of group isomorphisms and variety automorphisms. The findings offer a clearer characterization of algebraic groups and their mappings.

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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.