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Concept in mathematics
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates
Quaternionic_projective_space
Completion of the usual space with "points at infinity"
point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be
Projective_space
Type of topological space
Universal coefficient theorem. Complex projective space Quaternionic projective space Lens space Real projective plane See the table of Don Davis for a
Real_projective_space
Mathematical concept
inequality for complex projective space Projective Hilbert space Quaternionic projective space Real projective space Complex affine space K3 surface Besse,
Complex_projective_space
Concept in geometry
dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds. Marcel Berger's 1955
Quaternionic_manifold
conventional projective space to a point. More concretely, in a real projective space, complex projective space or quaternionic projective space K P n {\displaystyle
Stunted_projective_space
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
^{n+1}} (quaternionic n-space) and factor out by unit quaternion (= S 3 {\displaystyle S^{3}} ) multiplication to get the quaternionic projective space H P
Hopf_fibration
the corresponding Wolf space is the quaternionic projective space H P n {\displaystyle \mathbb {HP} _{n}} of (right) quaternionic lines through the origin
Quaternion-Kähler_manifold
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.) Second, the product of symmetric spaces is symmetric,
Simple_Lie_group
Geometric concept of a 2D space with "points at infinity" adjoined
the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space. Not all projective planes can
Projective_plane
Smooth manifold with an inner product on each tangent space
and real projective spaces with their standard metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley
Riemannian_manifold
subcategories. Euclidean space, Rn n-sphere, Sn n-torus, Tn Real projective space, RPn Complex projective space, CPn Quaternionic projective space, HPn Flag manifold
List_of_manifolds
Four-dimensional number system
Perotti, A. (2013). "Continuous slice functional calculus in quaternionic Hilbert spaces". Rev. Math. Phys. 25 (4): 1350006–126. arXiv:1207.0666. Bibcode:2013RvMaP
Quaternion
Type of topological space
Complex projective space CPn is a 2n-dimensional manifold. Quaternionic projective space HPn is a 4n-dimensional manifold. Manifolds related to projective space
Topological_manifold
Differential geometry concept
explains how one can associate a unique Wolf space to each of the simple complex Lie groups. Quaternionic discrete series representation Besse, Arthur
Quaternion-Kähler symmetric space
Quaternion-Kähler_symmetric_space
Topics referred to by the same term
England Higgs prime, H p n {\displaystyle Hp_{n}} HPN (gene) Quaternionic projective space, H P n {\displaystyle \mathbb {H} \mathrm {P} ^{n}} Westchester
HPN
Supergravity in eleven dimensions
7-sphere, which can be acquired by embedding the 7-sphere in a quaternionic projective space, with this giving a gauge group of SO ( 5 ) × SU ( 2 ) {\displaystyle
Eleven-dimensional supergravity
Eleven-dimensional_supergravity
(pseudo-)Riemannian manifold whose geodesics are reversible
Riemannian symmetric spaces. Basic examples of Riemannian symmetric spaces are Euclidean space, spheres, projective spaces, and hyperbolic spaces, each with their
Symmetric_space
Function theory with quaternion variable
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of
Quaternionic_analysis
Non-tensorial representation of the spin group
spinors in 3-dimensional Euclidean space are quaternionic, Weyl spinors in 4-dimensional Euclidean space are quaternionic, Weyl spinors in Lorentzian signature
Spinor
one symmetric spaces, together with real and quaternionic hyperbolic spaces, classification to which must be added one exceptional space, the Cayley plane
Complex_hyperbolic_space
Fiber bundle whose fibers are group torsors
is a principal S p ( 1 ) {\displaystyle Sp(1)} -bundle over quaternionic projective space H P n {\displaystyle \mathbb {H} \mathbb {P} ^{n}} . We then
Principal_bundle
Mathematical group
Paramodular group Projective unitary group Representations of classical Lie groups Symplectic manifold, Symplectic matrix, Symplectic vector space, Symplectic
Symplectic_group
Geometric model of the physical space
Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions
Three-dimensional_space
Algebraic structure designed for geometry
"The Grassmann method in projective geometry" A compilation of three notes on the application of exterior algebra to projective geometry C. Burali-Forti
Geometric_algebra
Optimal stable 2-systolic inequality
equality is attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on H P 2 {\displaystyle
Gromov's inequality for complex projective space
Gromov's_inequality_for_complex_projective_space
Special tangential structure
space BSp ( 1 ) ≅ BSU ( 2 ) {\displaystyle \operatorname {BSp} (1)\cong \operatorname {BSU} (2)} , which is the infinite quaternionic projective space
Spinh_structure
Relates the geometric vector bundles to algebraic projective modules
to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules are like vector bundles"
Serre–Swan_theorem
Vector bundle of rank 1
tautological line bundle on projective space. The projectivization P ( V ) {\displaystyle \mathbf {P} (V)} of a vector space V {\displaystyle V} over a
Line_bundle
Type of group in mathematics
traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups
Classical_group
Special type of principal bundle
is exactly the infinite quaternionic projective space H P ∞ {\displaystyle \mathbb {H} P^{\infty }} . For a topological space B {\displaystyle B} , let
Principal_SU(2)-bundle
Moduli space of the Yang–Mills equations
{M}}_{P}^{\mathrm {SD} }} diffeomorphic to a cone over the second complex projective space C P 2 {\displaystyle \mathbb {C} P^{2}} (whose tip corresponds to the
Yang–Mills_moduli_space
its normal spaces maps the manifold onto itself. Analogous Veronese embeddings are constructed for complex and quaternionic projective spaces, as well as
Veronese_map
Theorem in quantum mechanics
theorem to be applicable, the space on which measurements are defined must be a real or complex Hilbert space, or a quaternionic module. (Gleason's argument
Gleason's_theorem
Type of Riemannian manifold with constant Jacobi operator spectrum
{\displaystyle \mathbb {CH} ^{n}} , quaternionic projective spaces H P n {\displaystyle \mathbb {HP} ^{n}} , quaternionic hyperbolic spaces H H n {\displaystyle \mathbb
Osserman_manifold
Hypercomplex number system
basis with signature (− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element): ( I , j , k ) , ( i
Octonion
Manifold
variety Quaternionic manifold Real-complex manifold One must use the open unit ball in the C n {\displaystyle \mathbb {C} ^{n}} as the model space instead
Complex_manifold
Metric on a complex projective space endowed with Hermitian form
Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on a complex projective space CPn endowed with a Hermitian form. This metric was originally described
Fubini–Study_metric
Type of Riemannian manifold
respect to the Riemannian metric g {\displaystyle g} and satisfy the quaternionic relations I 2 = J 2 = K 2 = I J K = − 1 {\displaystyle I^{2}=J^{2}=K^{2}=IJK=-1}
Hyperkähler_manifold
Group of unitary matrices
as subgroup and the projective orthogonal group PO ( n ) {\displaystyle \operatorname {PO} (n)} as quotient, and the projective special orthogonal group
Unitary_group
Hypercomplex number system
e_{3}} , ..., e 15 {\displaystyle e_{15}} , which form a basis of the vector space of sedenions. Every sedenion can be represented in the form x = x 0 e 0
Sedenion
Generalized sphere of dimension n (mathematics)
Lie group structure Sp(1) = SU(2). 4-sphere Homeomorphic to the quaternionic projective line, H P 1 {\displaystyle \mathbf {HP} ^{1}} . SO ( 5 )
N-sphere
Form of differential geometry
the quaternionic projective plane is not its systolically optimal metric, in contrast with the 2-systole in the complex case. While the quaternionic projective
Systolic_geometry
Mathematical object
quaternion; that is, a quaternion that satisfies τ2 = −1. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie
3-sphere
American mathematician
exception of the real projective plane, which cannot be so embedded). This gave, for example, embedded 3-dimensional cones in Euclidean 4-space of every possible
H._Blaine_Lawson
Study of complex manifolds and several complex variables
otherwise. A projective complex analytic variety is a subset X ⊆ C P n {\displaystyle X\subseteq \mathbb {CP} ^{n}} of complex projective space that is, in
Complex_geometry
Mathematical operation
homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear
Cayley_transform
Element of a unital algebra over the field of real numbers
{\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yields a quaternionic matrix and its even subalgebra H ⊗ 2 ⊗ R C {\displaystyle \mathbb {H}
Hypercomplex_number
theory Projective geometry a form of geometry that studies geometric properties that are invariant under a projective transformation. Projective differential
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Particular projective representations of the orthogonal or special orthogonal groups
In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension
Spin_representation
element" (or a vector) is an old term for a Borel-weight vector. projective A projective representation of a group G is a group homomorphism π : G → P G
Glossary of representation theory
Glossary_of_representation_theory
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
sometimes denoted H(A,σ). 1. The set of self-adjoint real, complex, or quaternionic matrices with multiplication ( x y + y x ) / 2 {\displaystyle (xy+yx)/2}
Jordan_algebra
Equations describing classical electromagnetism
and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation was used. Maxwell's equations are partial differential equations
Maxwell's_equations
Topologist
Washington (September 1989). "Nonlinearly Equivalent Representations of Quaternionic 2-Groups" (PDF). Transactions of the American Mathematical Society. 315
Washington_Mio
Double cover Lie group of the special orthogonal group
Quotienting out by the entire center yields the minimal such group, the projective special orthogonal group, which is centerless, while quotienting out by
Spin_group
Mathematical concept
and Kottwitz (2005) Harry Reimann, The semi-simple zeta function of quaternionic Shimura varieties, Lecture Notes in Mathematics, 1657, Springer, 1997
Shimura_variety
American mathematician (1936–2023)
(2005), pp. 1504–1530. arXiv:math/0402283 Complex forms of quaternionic symmetric spaces, in Complex, contact and symmetric manifolds, Progress in Mathematics
Joseph_A._Wolf
Italian mathematician
2017) Quaternionic approximation: With application to slice regular functions (with Gal, Birkhäuser/Springer, 2019) Quaternionic de Branges spaces and characteristic
Irene_Sabadini
288-cell is the only non-regular 4-polytope which is the convex hull of a quaternionic group, disregarding the infinitely many dicyclic (same as binary dihedral)
Truncated_24-cells
Z2,0, repeated. KSp0(X) is the ring of stable equivalence classes of quaternionic vector bundles over X. Bott periodicity implies that the K-groups have
List_of_cohomology_theories
Ring homomorphism from the cobordism ring of manifolds to another ring
\epsilon \right)p_{1}^{3}\right]} Example (elliptic genus for quaternionic projective plane) : Φ e l l ( H P 2 ) = ∫ H P 2 1 90 [ ( − 4 δ 2 + 18 ϵ )
Genus of a multiplicative sequence
Genus_of_a_multiplicative_sequence
Quaternions with complex number coefficients
Complex Quaternions and Maxwell's Equations. Furey 2012. L. Silberstein, Quaternionic Form of Relativity, Philos. Mag. S., 6, Vol. 23, No. 137, pp. 790-809
Biquaternion
Quaternion of norm 1 (unit quaternion)
binary icosahedral group. A hyperbolic versor is a generalization of quaternionic versors to indefinite orthogonal groups, such as Lorentz group. It is
Versor
Concept in differential geometry
occurs as a holonomy group for locally symmetric spaces (that are locally isomorphic to the Cayley projective plane), and the second does not occur at all
Holonomy
Lie groups and their associated Lie algebras
symplectic matrices N 0 Z sp(2n,R) n(2n+1) Sp(n) compact symplectic group: quaternionic n×n unitary matrices Y 0 0 sp(n) n(2n+1) Mp(2n,R) metaplectic group:
Table_of_Lie_groups
Russian-French mathematician
Sobolev space theory. A sample application of Gromov and Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Matrix representing a Euclidean rotation
rotations, SO(3) is topologically equivalent to three-dimensional real projective space, RP3. Its universal covering group, Spin(3), is isomorphic to the 3-sphere
Rotation_matrix
Menger sponge Newton fractal Nova fractal - derived from Newton fractal Quaternionic fractal - three dimensional complex quadratic map Sierpinski carpet Sierpinski
List_of_chaotic_maps
Spin representations of the SO(3) group
can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose
Spinors_in_three_dimensions
Mathematical result in differential geometry
kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even.
Atiyah–Singer_index_theorem
Four-dimensional analog of the dodecahedron
S2CID 119288632. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8
120-cell
Development of linear transformations forming the Lorentz group
connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic
History of Lorentz transformations
History_of_Lorentz_transformations
Four-dimensional analog of the icosahedron
Cartesian coordinate — the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group
600-cell
American scientist (1839–1903)
other physicists of the convenience of the vectorial approach over the quaternionic calculus of William Rowan Hamilton, which was then widely used by British
Josiah_Willard_Gibbs
Generalization of a polytope in real space
triangular faces and 640 tetrahedral cells, seen in this 20-gonal projection. Quaternionic polytope Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation
Complex_polytope
German mathematician
geometry and quantum physics). In 1998 he was an Invited Speaker with talk Quaternionic analysis of Riemann surfaces and differential geometry at the International
Ulrich_Pinkall
Low-rank isomorphisms in mathematics
encoded in the incidence relation between spacetime points and projective lines in twistor space. In this way, the exceptional isomorphism is built into the
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
Italian mathematician (1911–1999)
; Pontecorvo, M., eds. (1999), Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics. Dedicated to the Memory of André
Enzo_Martinelli
researcher Katrin Leschke (born 1968), German differential geometer, quaternionic analyst, and minimal surface theorist Nandi Olive Leslie, American industrial
List_of_women_in_mathematics
American mathematician
MR 0133834. Floyd, E. E. (1971). "Stiefel-Whitney numbers of quaternionic and related manifolds". Trans. Amer. Math. Soc. 155: 77–94. doi:10
Edwin_E._Floyd
QUATERNIONIC PROJECTIVE-SPACE
QUATERNIONIC PROJECTIVE-SPACE
Boy/Male
Greek
Productive.
Girl/Female
Muslim
Protective Angel
Girl/Female
Muslim/Islamic
Protective angel
Girl/Female
Muslim
Protective Angel
Boy/Male
German
Protective
Girl/Female
Irish
Protective.
Boy/Male
Christian & English(British/American/Australian)
Protective Friend
Girl/Female
Irish
Protective.
Biblical
a guard of four soldiers,...and delivered him to four quaternions of soldiers to guard him...
Boy/Male
Arabic, Indian, Muslim, Sindhi
Protective; Safety
Girl/Female
Indian
Protective Angel
Girl/Female
Celtic, French, German, Irish
Strong; Protective
Girl/Female
German, Swedish
Protective Victory
Girl/Female
Indian
Protective Angel
Girl/Female
German American
Protective.
Boy/Male
Christian & English(British/American/Australian)
Protective Grace
Boy/Male
Polish
Protective shield.
Boy/Male
British, English, Netherlands
Protective
Boy/Male
German
Protective
Girl/Female
Muslim/Islamic
Protective angel
QUATERNIONIC PROJECTIVE-SPACE
QUATERNIONIC PROJECTIVE-SPACE
Boy/Male
Celtic American Welsh
Gray.
Surname or Lastname
English
English : unexplained; possibly a variant spelling of Vial. Compare Viles.
Boy/Male
Hindu
Boy/Male
Australian, Russian, Ukrainian
Little; Form of Paul; Small
Boy/Male
Gujarati, Hindu, Indian, Kannada, Marathi, Oriya, Telugu
Pari
Girl/Female
Indian
Prayer, Supplication
Boy/Male
Anglo, Australian, British, Danish, English, French, German
Ruler; Rough Land
Boy/Male
Muslim
Civilized
Male
English
 Short form of English Lewis, LEW means "famous warrior." Compare with another form of Lew.
Girl/Female
Hindu
Confirmation, Healthy, Possessor of all wealth, Healthy, Possessor of all wealth, Nourishment, Endorsement
QUATERNIONIC PROJECTIVE-SPACE
QUATERNIONIC PROJECTIVE-SPACE
QUATERNIONIC PROJECTIVE-SPACE
QUATERNIONIC PROJECTIVE-SPACE
QUATERNIONIC PROJECTIVE-SPACE
n.
A part of mechanics which treats of the motion, range, time of flight, etc., of bodies thrown or driven through the air by an impelling force.
n.
The quotient of two vectors, or of two directed right lines in space, considered as depending on four geometrical elements, and as expressible by an algebraic symbol of quadrinomial form.
n.
A perspective glass.
a.
Pertaining to projection, or to a projectile.
a.
Bringing into being; causing to exist; producing; originative; as, an age productive of great men; a spirit productive of heroic achievements.
n.
The scene before or around, in time or in space; view; prospect.
a.
Affording protection; sheltering; defensive.
n.
The quality or state of projecting, or being projected; projection; protrusion.
n.
A word of four syllables; a quadrisyllable.
n.
Being within view or consideration, as a future event or contingency; relating to the future: expected; as, a prospective benefit.
n.
The turning factor of a quaternion.
a.
Caused or imparted by impulse or projection; impelled forward; as, projectile motion.
n.
The number four.
n.
The representation of something; delineation; plan; especially, the representation of any object on a perspective plane, or such a delineation as would result were the chief points of the object thrown forward upon the plane, each in the direction of a line drawn through it from a given point of sight, or central point; as, the projection of a sphere. The several kinds of projection differ according to the assumed point of sight and plane of projection in each.
n.
A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.
a.
Projecting or impelling forward; as, a projectile force.
v. t.
To divide into quaternions, files, or companies.
n.
The number four; a collection of four things; a quaternion.
n.
A body projected, or impelled forward, by force; especially, a missile adapted to be shot from a firearm.
a.
Having the quality or power of producing; yielding or furnishing results; as, productive soil; productive enterprises; productive labor, that which increases the number or amount of products.