Det
edit- Dieudonné det etc. https://core.ac.uk/download/pdf/82261662.pdf
- F.F. Knudsen and D. Mumford,The projectivity of the moduli space of stable curves I.Preliminaries on “det” and “Div”, Math. Scand.39(1976), no. 1, 19–55.
- [1]
- Dodgson condensation - The theory of determinants in the historical order of development Volume 3 by Thomas Muir, p. 17
- Chio condensation, Cramers rule https://hal.archives-ouvertes.fr/hal-01500199/file/HA.pdf
- The Linear Algebra a Beginning Graduate Student Ought to Know by Jonathan Samuel Golan, p 208
- [2]
Duality notes
editRiesz–Markov–Kakutani representation theorem: linear functionals on a space vs. measures.
quadratic reciprocity, cubic reciprocity
duality | explanation, context | conceptual explanation | application | generalization | |
---|---|---|---|---|---|
Schur–Weyl duality | representations of symmetric group and GLn | see A categorical approach to classical and quantumSchur–Weyl duality http://www.maths.usyd.edu.au/u/pubs/publist/preprints/2010/davydov-17.pdf ; Double centralizer theorem, Thm 4.54 in http://math.mit.edu/~etingof/replect.pdf,: Let A, B be two subalgebras of the algebra End E of endomorphisms of a finite dimensional vector space E, such that A is semisimple, and . Then , more... | ?? | ||
Spanier–Whitehead duality (also called S-duality (homotopy theory)) | embed a topological space into some S^n, stable homotopy type; | Dold: "Fixed point index and fixed point theorem for Euclidean neighborhood retracts"; explained in Ponto; Shulman "Traces in symmetric monoidal categories" | example: Alexander duality | ||
Stone's duality | restricts from an adjunction to sober spaces and spatial locales | Example: Duality theory for distributive lattices, gives rise to spectral spaces | |||
Tannaka–Krein duality | generalization of Pontryagin for nonabelian groups, . reconstruct a group from its unitary representations | equivalence of categories? | |||
Pontryagin duality | dualizing object, Hom(-, S^1); can also be proven by Poincaré-Lefschetz duality and topological means, see http://www.encyclopediaofmath.org/index.php/Pontryagin_duality | Fourier transform (case R^n) in harmonic analysis; reciprocal (or dual) lattice in crystallography ( , dual lattice . | Tannaka-Krein; variant for group schemes Cartier duality: Z/p is dual to \mu_p | ||
Dual pair | vector spaces V x W \r K, | example (by Hahn-Banach) locally convex topological vector space; duality between Lp spaces; Riesz representation theorem: identifies a Hilbert space with its dual.; Convex conjugate: f sometimes = f**, described by Fenchel–Moreau theorem, basis of duality (optimization), in general only weak duality | |||
Dual curve | tangents to a curve in P^2; curve in dual projective plane | Legendre transform, see
Arnold, Vladimir Igorevich (1988), Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, ISBN 3-540-96649-8 | |||
dual space | vector spaces or topological vector space | dual object (also called dualizable object) | other examples: dual bundle, distribution (mathematics) | ||
dualizing sheaf | algebraic geometry | Adjunction formula | |||
Poincaré duality | Twisted Poincaré duality (local coefficients), Lefschetz duality (manifolds with boundary), Verdier duality for singularities | ||||
Coherent duality | chain of adjoints, see Balmer, Dell'Ambrogio, Sanders. Grothendieck-Neeman duality and the Wirthmüller isomorphism | Serre duality, hence Riemann-Roch theorem | |||
Verdier duality | chain of adjoints as in Coherent duality | ||||
dual object | examples: Dual representation | ||||
Hopf algebras vs. group schemes | equivalence of categories | ||||
Dual abelian variety | equivalence of categories | ||||
dual cone | adjunction between the category of subsets of R^n; C** is the closure of the smallest convex cone containing C; equivalence on the subcategory of closed convex cones. | toric geometry, e.g. dual of <e_2, 2e_1-e_2> spanned by <f_1, f_1 + f_2, f_1+2 f_2>, hence C[U,V,W]/(V^2-UW), see Fulton | |||
- Alvis–Curtis duality
- Araki duality
- Beta-dual space
- Dual basis in a field extension
- Dual (category theory)
- Dual graph
- Dual polygon
- Dual polyhedron
- Dual problem
- Dual q-Hahn polynomials
- Dual q-Krawtchouk polynomials
- Dual topology: topology on a pair of topologically dual locally convex vector spaces
- Dual wavelet
- Duality (optimization), also called Lagrange duality
- Duality (order theory)
- Duality of stereotype spaces
- Duality theory for distributive lattices
- Dualizing complex
- Eckmann–Hilton duality
- Esakia duality
- Fenchel's duality theorem
- Haag duality
- Jónsson–Tarski duality
- Langlands dual
- Local Tate duality
- Opposite category
- Poitou–Tate duality
- Morita duality? [3]
- duality in measure theory ? [4]
Coactions may be thought of as "actions of the dual group even when there isn't any dual group [5]