20Gxx Linear algebraic groups
This subtopic studies linear algebraic groups, where group structure is studied through algebraic varieties and matrix representations.
Specific topics
20G05 Representation theory of linear algebraic groups
Overview
20G05 studies representation theory of linear algebraic groups in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for representation theory of linear algebraic groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G07 Structure theory for linear algebraic groups
Overview
20G07 studies structure theory for linear algebraic groups in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for structure theory for linear algebraic groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G10 Cohomology theory for linear algebraic groups
Overview
20G10 studies cohomology theory for linear algebraic groups in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for cohomology theory for linear algebraic groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G15 Linear algebraic groups over arbitrary fields
Overview
20G15 studies linear algebraic groups over arbitrary fields in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear algebraic groups over arbitrary fields
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
Overview
20G20 studies linear algebraic groups over the reals, the complexes, the quaternions in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear algebraic groups over the reals, the complexes, the quaternions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G25 Linear algebraic groups over local fields and their integers
Overview
20G25 studies linear algebraic groups over local fields and their integers in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear algebraic groups over local fields and their integers
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G30 Linear algebraic groups over global fields and their integers
Overview
20G30 studies linear algebraic groups over global fields and their integers in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear algebraic groups over global fields and their integers
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G35 Linear algebraic groups over adèles and other rings and schemes
Overview
20G35 studies linear algebraic groups over adeles and other rings and schemes in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear algebraic groups over adeles and other rings and schemes
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G40 Linear algebraic groups over finite fields
Overview
20G40 studies linear algebraic groups over finite fields in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear algebraic groups over finite fields
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G41 Exceptional groups
Overview
20G41 studies exceptional groups in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for exceptional groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G42 Quantum groups (algebraic aspects)
Overview
20G42 studies quantum groups (algebraic aspects) in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for quantum groups (algebraic aspects)
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G43 Schur and $q$-Schur algebras
Overview
20G43 studies schur and $q$-schur algebras in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for schur and $q$-schur algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G44 Kac-Moody groups
Overview
20G44 studies kac-moody groups in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for kac-moody groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks
20G45 Applications of linear algebraic groups to the sciences
Overview
20G45 studies applications of linear algebraic groups to the sciences in linear algebraic groups. It studies groups defined by polynomial equations and their structural, cohomological, and representation-theoretic properties.
Related Wikipedia Page
Linear Algebraic Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of linear algebraic groups to the sciences
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in algebraic geometry, number theory, and the theory of algebraic symmetry.
Applications
- Structure and cohomology
- Fields and arithmetic of algebraic groups
- Applications to geometry and representation theory
References
Recommended Textbooks