17Bxx Lie algebras and Lie superalgebras
This subtopic studies Lie algebras and Lie superalgebras, including structure, representations, and links to geometry, differential equations, and physics.
Specific topics
17B01 Identities, free Lie (super)algebras
Overview
17B01 studies structure theory for lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Structure theory for Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for structure theory for lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B05 Structure theory of Lie algebras
Overview
17B05 studies nilpotent and solvable lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Nilpotent and solvable Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for nilpotent and solvable lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B08 Coadjoint orbits; nilpotent varieties
Overview
17B08 studies lie algebras associated with algebraic groups in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Lie algebras associated with algebraic groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie algebras associated with 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B10 Representations, algebraic theory
Overview
17B10 studies semisimple lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Semisimple Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for semisimple lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B15 Representations, analytic theory
Overview
17B15 studies representation theory of lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Representation theory of Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for representation theory of lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B20 Simple, semisimple, reductive Lie algebras
Overview
17B20 studies finite-dimensional lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Finite-dimensional Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for finite-dimensional lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B22 Root systems
Overview
17B22 studies root systems and weyl groups in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Root systems and Weyl groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for root systems and weyl 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B25 Exceptional Lie algebras
Overview
17B25 studies infinite-dimensional lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Infinite-dimensional Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for infinite-dimensional lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B30 Solvable, nilpotent Lie algebras
Overview
17B30 studies lie algebra cohomology in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Lie algebra cohomology (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie algebra cohomology
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B35 Universal enveloping algebras and related constructions
Overview
17B35 studies cohomology of lie superalgebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Cohomology of Lie superalgebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for cohomology of lie superalgebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B37 Quantum groups and related deformations
Overview
17B37 studies lie superalgebras and graded structures in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Lie superalgebras and graded structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie superalgebras and graded structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B40 Automorphisms, derivations, other operators for Lie algebras
Overview
17B40 studies hamiltonian, poisson, and symplectic lie structures in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Hamiltonian, Poisson, and symplectic Lie structures (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for hamiltonian, poisson, and symplectic lie structures
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B45 Lie algebras of linear algebraic groups
Overview
17B45 studies deformations and quantum groups related to lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Deformations and quantum groups related to Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for deformations and quantum groups related to lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B50 Modular Lie algebras
Overview
17B50 studies applications of lie algebras in differential equations and geometry in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Applications of Lie algebras in differential equations and geometry (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of lie algebras in differential equations and geometry
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B55 Homological methods in Lie algebras
Overview
17B55 studies affine and kac-moody lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Affine and Kac-Moody Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for affine and kac-moody lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B56 Cohomology of Lie algebras
Overview
17B56 studies extended affine lie algebras and lie tori in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Extended affine Lie algebras and Lie tori (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for extended affine lie algebras and lie tori
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B60 Lie algebras: applications to physics
Overview
17B60 studies lie algebra automorphisms, derivations, and extensions in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Lie algebra automorphisms, derivations, and extensions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie algebra automorphisms, derivations, and extensions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B63 Poisson algebras
Overview
17B63 studies lie algebra invariants and identities in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Lie algebra invariants and identities (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie algebra invariants and identities
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B65 Infinite-dimensional Lie algebras
Overview
17B65 studies computational methods for lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Computational methods for Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for computational methods for lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B66 Lie algebras of vector fields
Overview
17B66 studies lie algebra representations over fields of positive characteristic in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Lie algebra representations over fields of positive characteristic (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie algebra representations over fields of positive characteristic
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B67 Kac-Moody algebras
Overview
17B67 studies restricted lie algebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Restricted Lie algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for restricted lie 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 geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B68 Virasoro and related algebras
Overview
17B68 studies modular lie algebras and representation theory in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Modular Lie algebras and representation theory (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for modular lie algebras and representation theory
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B69 Vertex operators; vertex operator algebras
Overview
17B69 studies infinite-dimensional lie superalgebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Infinite-dimensional Lie superalgebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for infinite-dimensional lie superalgebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B70 Graded Lie algebras
Overview
17B70 studies lie superalgebra representations in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Lie superalgebra representations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie superalgebra representations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B75 Color Lie superalgebras, graded Lie superalgebras
Overview
17B75 studies applications of lie superalgebras in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Applications of Lie superalgebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of lie superalgebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B80 Applications of Lie algebras and superalgebras to integrable systems
Overview
17B80 studies miscellaneous lie algebra topics in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Miscellaneous Lie algebra topics (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous lie algebra topics
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks
17B81 Applications to physics
Overview
17B81 studies miscellaneous lie superalgebra topics in Lie algebras and Lie superalgebras. It studies bracketed algebraic structures governing symmetry, representation theory, and graded or super-graded extensions.
Related Wikipedia Page
Miscellaneous Lie superalgebra topics (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous lie superalgebra topics
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in geometry, mathematical physics, representation theory, and the classification of symmetries.
Applications
- Symmetry and representation theory
- Graded and super-graded Lie structures
- Applications to geometry and physics
References
Recommended Textbooks