Abelians

Abelians

Abelians refers to concepts and entities related to the mathematician Niels Henrik Abel and his contributions to various fields in mathematics. The term is often used in the context of Abelian groups, Abelian varieties, and other mathematical structures that exhibit commutative properties.

Abelian Groups

An Abelian group is a group in which the group operation is commutative. This means that for any two elements a and b in the group, the equation a * b = b * a holds. Abelian groups are named after Niels Henrik Abel, who made significant contributions to the field of group theory.

Abelian Varieties

An Abelian variety is a complete algebraic variety that has a group structure and is projective. These varieties are important in number theory and algebraic geometry. They generalize the concept of an elliptic curve and are used in various areas of mathematics, including the study of modular forms and Galois representations.

Abelian Categories

An Abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. Abelian categories are a central concept in homological algebra and are used to study chain complexes and cohomology theories.

Abelian Functions

Abelian functions are a generalization of elliptic functions and are related to the inversion of Abelian integrals. These functions play a significant role in the theory of complex analysis and algebraic geometry.

Related Concepts

• Commutative algebra
• Homological algebra
• Elliptic curve
• Modular form
• Galois representation
• Complex analysis
• Algebraic geometry

See Also

• Niels Henrik Abel
• Group theory
• Elliptic function
• Category theory
• Homological algebra
• Number theory