17Axx General nonassociative rings and algebras
This subtopic studies general nonassociative rings and algebras, developing foundational concepts and structural tools for systems where associativity is relaxed.
Specific topics
17A01 General theory of nonassociative rings and algebras
Overview
17A01 studies foundations of nonassociative algebra in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Foundations of nonassociative algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for foundations of nonassociative algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A05 Power-associative rings
Overview
17A05 studies power-associative and alternative algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Power-associative and alternative algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for power-associative and alternative algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A15 Noncommutative Jordan algebras
Overview
17A15 studies flexible and jordan-adjacent identities in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Flexible and Jordan-adjacent identities (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for flexible and jordan-adjacent identities
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A20 Flexible algebras
Overview
17A20 studies lie-admissible and jordan-admissible algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Lie-admissible and Jordan-admissible algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for lie-admissible and jordan-admissible algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A30 Algebras satisfying other identities
Overview
17A30 studies nuclear and alternative identities in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Nuclear and alternative identities (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for nuclear and alternative identities
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A32 Leibniz algebras
Overview
17A32 studies polynomial identities in nonassociative algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Polynomial identities in nonassociative algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for polynomial identities in nonassociative algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A35 Division algebras
Overview
17A35 studies symmetric and anti-symmetric constructions in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Symmetric and anti-symmetric constructions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for symmetric and anti-symmetric constructions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A36 Automorphisms, derivations, other operators for nonassociative algebras
Overview
17A36 studies cohomology and deformation of nonassociative algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Cohomology and deformation of nonassociative algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for cohomology and deformation of nonassociative algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A40 Ternary compositions
Overview
17A40 studies structure theory of nonassociative algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Structure theory of nonassociative algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for structure theory of nonassociative algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A42 Other $n$-ary compositions ($n \geq 3$)
Overview
17A42 studies automorphisms and derivations in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Automorphisms and derivations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for automorphisms and derivations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A45 Quadratic algebras
Overview
17A45 studies representations and modules over nonassociative algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Representations and modules over nonassociative algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for representations and modules over nonassociative algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A50 Free nonassociative algebras
Overview
17A50 studies classification problems in nonassociative algebra in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Classification problems in nonassociative algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for classification problems in nonassociative algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A60 Structure theory for nonassociative algebras
Overview
17A60 studies applications of nonassociative algebra in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Applications of nonassociative algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of nonassociative algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A65 Radical theory for nonassociative algebras
Overview
17A65 studies computational methods in nonassociative algebra in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Computational methods in nonassociative algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for computational methods in nonassociative algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A70 Superalgebras
Overview
17A70 studies special classes of nonassociative algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Special classes of nonassociative algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for special classes of nonassociative algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A75 Composition algebras
Overview
17A75 studies identities and varieties of nonassociative algebras in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Identities and varieties of nonassociative algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for identities and varieties of nonassociative algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks
17A80 Valued algebras
Overview
17A80 studies miscellaneous topics in nonassociative algebra in general nonassociative rings and algebras. It studies algebraic systems where associativity is relaxed, with emphasis on identities, structural decomposition, and representation-theoretic behavior.
Related Wikipedia Page
Miscellaneous topics in nonassociative algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous topics in nonassociative algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to classify nonassociative multiplication rules and to study algebras arising from geometry and physics.
Applications
- Alternative, Jordan, and Lie-admissible structures
- Classification and representation problems
- Computational and deformation methods
References
Recommended Textbooks