Оглавление
Abdenacer Makhlouf. Algebra and Applications 1
Table of Contents
List of Illustrations
Guide
Pages
Algebra and Applications 1. Non-associative Algebras and Categories
Foreword
1. Jordan Superalgebras
1.1. Introduction
1.2. Tits–Kantor–Koecher construction
1.3. Basic examples (classical superalgebras)
1.4. Brackets
1.5. Cheng–Kac superalgebras
1.6. Finite dimensional simple Jordan superalgebras. 1.6.1. Case F is algebraically closed and char F = 0
1.6.2. Case char F = p > 2, the even partis semisimple
1.6.3. Case char F = p> 2, the even partis not semisimple
1.6.4. Non-unital simple Jordan superalgebras
1.7. Finite dimensional representations
1.7.1. Superalgebras of rank ≥ 3
1.7.2. Superalgebras of rank ≤ 2
1.8. Jordan superconformal algebras
1.9. References
2. Composition Algebras
2.1. Introduction
2.2. Quaternions and octonions
2.2.1. Quaternions
2.2.2. Rotations in three- (and four-) dimensional space
2.2.3. Octonions
2.3. Unital composition algebras
2.3.1. The Cayley–Dickson doubling process and the generalized Hurwitz theorem
2.3.2. Isotropic Hurwitz algebras
2.4. Symmetric composition algebras
2.5. Triality
2.6. Concluding remarks
2.7. Acknowledgments
2.8. References
3. Graded-Division Algebras
3.1. Introduction
3.2. Background on gradings
3.2.1. Gradings induced by a group homomorphism
3.2.2. Weak isomorphism and equivalence
3.2.3. Basic properties of division gradings
3.2.4. Graded presentations of associative algebras
3.2.4.1. Power norm residue symbols
3.2.4.2. Clifford gradings
3.2.5. Tensor products of division gradings
3.2.6. Loop construction. 3.2.6.1. Centroid of a graded algebra
3.2.6.2. Loop algebras with a split centroid
3.2.7. Another construction of graded-simple algebras
3.3. Graded-division algebras over algebraically closed fields
3.4. Real graded-division associative algebras
3.4.1. Simple graded-division algebras
3.4.2. Pauli gradings
3.4.3. Commutative case
3.4.4. Non-commutative graded-division algebras with one-dimensional homogeneous components
3.4.5. Equivalence classes of graded-division algebras with one-dimensional homogeneous components. 3.4.5.1. Basic algebras
3.4.5.2. Tensor products of basic algebras: commutation relations
3.4.5.3. Tensor products of basic algebras: equivalence
3.4.5.4. Tensor products of basic algebras: non-equivalence
3.4.6. Graded-division algebras with non-central two-dimensional identity components
3.4.6.1. Equivalence of graded-division algebras with non-central two-dimensional identity components
3.4.7. Graded-division algebras with four-dimensional identity components
3.4.8. Classification of real graded-division algebras, up to isomorphism
3.5. Real loop algebras with a non-split centroid
3.6. Alternative algebras
3.6.1. Cayley–Dickson doubling process
3.6.2. Gradings on octonion algebras
3.6.3. Graded-simple real alternative algebras
3.6.4. Graded-division real alternative algebras
3.7. Gradings of fields
3.8. References
4. Non-associative C*-algebras
4.1. Introduction
4.2. JB-algebras
4.3. The non-associative Vidav–Palmer and Gelfand–Naimark theorems
4.4. JB*-triples
4.5. Past, present and future of non-associative C*-algebras
4.6. Acknowledgments
4.7. References
5. Structure of H*-algebras
5.1. Introduction
5.2. Preliminaries: aspects of the general theory
5.3. Ultraproducts of H*-algebras
5.4. Quadratic H*-algebras
5.5. Associative H*-algebras
5.6. Flexible H*-algebras
5.7. Non-commutative Jordan H*-algebras
5.8. Jordan H*-algebras
5.9. Moufang H*-algebras
5.10. Lie H*-algebras
5.11. Topics closely related to Lie H*-algebras
5.12. Two-graded H*-algebras
5.13. Other topics: beyond the H*-algebras
5.14. Acknowledgments
5.15. References
6. Krichever–Novikov Type Algebras: Definitions and Results
6.1. Introduction
6.2. The Virasoro algebra and its relatives
6.3. The geometric picture
6.3.1. The geometric realizations of the Witt algebra
6.3.2. Arbitrary genus generalizations
6.3.3. Meromorphic forms
6.4. Algebraic structures
6.4.1. Associative structure
6.4.2. Lie and Poisson algebra structure
6.4.3. The vector field algebra and the Lie derivative
6.4.4. The algebra of differential operators
6.4.5. Differential operators of all degrees
6.4.6. Lie superalgebras of half forms
6.4.7. Jordan superalgebra
6.4.8. Higher genus current algebras
6.4.9. KN-type algebras
6.5. Almost-graded structure. 6.5.1. Definition of almost-gradedness
6.5.2. Separating cycle and KN pairing
6.5.3. The homogeneous subspaces
6.5.4. The algebras
6.5.5. Triangular decomposition and filtrations
6.6. Central extensions
6.6.1. Central extensions and cocycles
6.6.2. Geometric cocycles
6.6.2.1. Projective and affine connections
6.6.2.2. The function algebra
6.6.2.3. The current algebra
6.6.2.4. The vector field algebra
6.6.2.5. The differential operator algebra
6.6.2.6. The Lie superalgebra
6.6.3. Uniqueness and classification of central extensions
6.7. Examples and generalizations. 6.7.1. The genus zero and three-point situation
6.7.2. Genus zero multipoint algebras – integrable systems
6.7.3. Deformations
6.8. Lax operator algebras
6.9. Fermionic Fock space. 6.9.1. Semi-infinite forms and fermionic Fock space representations
6.9.2. b–csystems
6.10. Sugawara representation
6.11. Application to moduli space
6.12. Acknowledgments
6.13. References
7. An Introduction to Pre-Lie Algebras
7.1. Introduction
7.1.1. Explanation of notions
7.1.2. Two fundamental properties
7.1.3. Some subclasses
7.1.4. Organization of this chapter
7.2. Some appearances of pre-Lie algebras
7.2.1. Left-invariant affine structures on Lie groups: a geometric interpretation of “left-symmetry”
7.2.2. Deformation complexes of algebras and right-symmetric algebras
7.2.3. Rooted tree algebras: free pre-Lie algebras
7.2.4. Complex structures on Lie algebras
7.2.5. Symplectic structures on Lie groups and Lie algebras, phase spaces of Lie algebras and Kähler structures
7.2.6. Vertex algebras
7.3. Some basic results and constructions of pre-Lie algebras
7.3.1. Some basic results of pre-Lie algebras. 7.3.1.1. Some studies on structure theory
7.3.1.2. Some comments on representation theory
7.3.1.3. Some classification results
7.3.1.4. Summary: main problems and ideas
7.3.2. Constructions of pre-Lie algebras from some known structures. 7.3.2.1. Constructions from commutative associative algebras
7.3.2.2. Constructions from Lie algebras
7.3.2.3. Constructions from associative algebras
7.3.2.4. Constructions from linear functions
7.4. Pre-Lie algebras and CYBE
7.4.1. The existence of a compatible pre-Lie algebra on a Lie algebra
7.4.2. CYBE: unification of tensor and operator forms
7.4.3. Pre-Lie algebras,-operators and CYBE
7.4.4. An algebraic interpretation of “left-symmetry”: construction from Lie algebras revisited
7.5. A larger framework: Lie analogues of Loday algebras
7.5.1. Pre-Lie algebras, dendriform algebras and Loday algebras
7.5.2. L-dendriform algebras
7.5.3. Lie analogues of Loday algebras
7.5.3.1. “Rule” of construction
7.6. References
8. Symplectic, Product and Complex Structures on 3-Lie Algebras
8.1. Introduction
8.2. Preliminaries
8.3. Representations of 3-pre-Lie algebras
8.4. Symplectic structures and phase spaces of 3-Lie algebras
8.5. Product structures on 3-Lie algebras
8.6. Complex structures on 3-Lie algebras
8.7. Complex product structures on 3-Lie algebras
8.8. Para-Kähler structures on 3-Lie algebras
8.9. Pseudo-Kähler structures on 3-Lie algebras
8.10. References
9. Derived Categories
9.1. Introduction
9.2. Grothendieck’s definition
9.3. Verdier’s definition
9.4. Triangulated structure
9.5. Derived functors
9.6. Derived Morita theory
9.7. Dg categories
9.7.1. Dg categories and functors
9.7.2. The derived category
9.7.3. Derived functors
9.7.4. Dg quotients
9.7.5. Invariants
9.8. References
List of Authors
Index
A
B, C
D
E, F
G
H
I, J
K, L
M, N
O, P
Q, R
S
T
U, V
W
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