16Txx Hopf algebras and quantum groups
This subtopic studies Hopf algebras and quantum groups, emphasizing algebraic symmetries, duality, and interactions with geometry, topology, and mathematical physics.
Specific topics
16T05 Hopf algebras and their applications
Overview
16T05 studies hopf algebras, their structure and classification in Hopf algebras and quantum groups. It studies algebraic systems with compatible multiplication, comultiplication, and antipode operations that model symmetry in algebra, geometry, and mathematical physics.
Related Wikipedia Page
Hopf algebras, their structure and classification (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for hopf algebras, their structure and classification
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize symmetry actions, tensor-compatible algebraic constructions, and deformation-theoretic methods.
Applications
- Symmetry and deformation in algebra
- Tensor-categorical methods
- Applications to noncommutative geometry and physics
References
Recommended Textbooks
16T10 Bialgebras
Overview
16T10 studies quantum groups and related deformations in Hopf algebras and quantum groups. It studies algebraic systems with compatible multiplication, comultiplication, and antipode operations that model symmetry in algebra, geometry, and mathematical physics.
Related Wikipedia Page
Quantum groups and related deformations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for quantum groups and related deformations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize symmetry actions, tensor-compatible algebraic constructions, and deformation-theoretic methods.
Applications
- Symmetry and deformation in algebra
- Tensor-categorical methods
- Applications to noncommutative geometry and physics
References
Recommended Textbooks
16T15 Coalgebras and comodules
Overview
16T15 studies hopf actions and smash products in Hopf algebras and quantum groups. It studies algebraic systems with compatible multiplication, comultiplication, and antipode operations that model symmetry in algebra, geometry, and mathematical physics.
Related Wikipedia Page
Hopf actions and smash products (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for hopf actions and smash products
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize symmetry actions, tensor-compatible algebraic constructions, and deformation-theoretic methods.
Applications
- Symmetry and deformation in algebra
- Tensor-categorical methods
- Applications to noncommutative geometry and physics
References
Recommended Textbooks
16T20 Ring-theoretic aspects of quantum groups
Overview
16T20 studies coalgebras and comodules in Hopf algebras and quantum groups. It studies algebraic systems with compatible multiplication, comultiplication, and antipode operations that model symmetry in algebra, geometry, and mathematical physics.
Related Wikipedia Page
Coalgebras and comodules (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for coalgebras and comodules
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize symmetry actions, tensor-compatible algebraic constructions, and deformation-theoretic methods.
Applications
- Symmetry and deformation in algebra
- Tensor-categorical methods
- Applications to noncommutative geometry and physics
References
Recommended Textbooks
16T25 Yang-Baxter equations
Overview
16T25 studies bialgebras and quasitriangular structures in Hopf algebras and quantum groups. It studies algebraic systems with compatible multiplication, comultiplication, and antipode operations that model symmetry in algebra, geometry, and mathematical physics.
Related Wikipedia Page
Bialgebras and quasitriangular structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for bialgebras and quasitriangular structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize symmetry actions, tensor-compatible algebraic constructions, and deformation-theoretic methods.
Applications
- Symmetry and deformation in algebra
- Tensor-categorical methods
- Applications to noncommutative geometry and physics
References
Recommended Textbooks
16T30 Connections of Hopf algebras with combinatorics
Overview
16T30 studies quantum symmetries and applications in Hopf algebras and quantum groups. It studies algebraic systems with compatible multiplication, comultiplication, and antipode operations that model symmetry in algebra, geometry, and mathematical physics.
Related Wikipedia Page
Quantum symmetries and applications (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for quantum symmetries and applications
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize symmetry actions, tensor-compatible algebraic constructions, and deformation-theoretic methods.
Applications
- Symmetry and deformation in algebra
- Tensor-categorical methods
- Applications to noncommutative geometry and physics
References
Recommended Textbooks