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Representation of a group or algebra in terms of an algebra with quaternionic structure
field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure
Quaternionic_representation
mathematics, a quaternionic discrete series representation is a discrete series representation of a semisimple Lie group G associated with a quaternionic structure
Quaternionic discrete series representation
Quaternionic_discrete_series_representation
Four-dimensional number system
Quaternionic manifold – Concept in geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective
Quaternion
v+W\mapsto gv+W} . quaternionic A quaternionic representation of a group G is a complex representation equipped with a G-invariant quaternionic structure. quiver
Glossary of representation theory
Glossary_of_representation_theory
irreducible complex representation of G with Schur indicator −1, called a quaternionic representation. Moreover, every irreducible representation on a complex
Frobenius–Schur_indicator
Particular projective representations of the orthogonal or special orthogonal groups
that the triple i, j and k:=ij make S into a quaternionic vector space SH. This is called a quaternionic structure. There is an invariant complex antilinear
Spin_representation
argument), one can show that any complex symplectic representation is a quaternionic representation. Quaternionic representations of finite or compact groups
Symplectic_representation
Concept in linear algebra
A quaternionic matrix is a matrix whose elements are quaternions. The quaternions form a noncommutative ring, and therefore addition and multiplication
Quaternionic_matrix
term complex representation is reserved for a representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words
Complex_representation
Representations of finite groups, particularly on vector spaces
representations of G . {\displaystyle G.} Definition. A quaternionic representation is a (complex) representation V , {\displaystyle V,} which possesses a G {\displaystyle
Representation theory of finite groups
Representation_theory_of_finite_groups
representation Semisimple Complex representation Real representation Quaternionic representation Pseudo-real representation Symplectic representation
List of representation theory topics
List_of_representation_theory_topics
Type of representation in representation theory
pseudoreal representation. An irreducible pseudoreal representation V is necessarily a quaternionic representation: it admits an invariant quaternionic structure
Real_representation
Non-tensorial representation of the spin group
of the representation; this is the algebraic origin of Majorana conditions. When S {\displaystyle S} is of quaternionic type, the representation carries
Spinor
Mathematical group
\operatorname {Sp} (n)} is given by the quaternionic skew-Hermitian matrices, the set of n × n {\displaystyle n\times n} quaternionic matrices that satisfy A + A
Symplectic_group
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
Polyhedron with 9 faces
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011), "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8 {\displaystyle
Enneahedron
63rd Johnson solid (8 faces)
Koca, Mehmet; Al-Ajmi, Mudhahir; Koca, Nazife Ozdes (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8 {\displaystyle
Tridiminished_icosahedron
Four-dimensional analog of the dodecahedron
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system"
120-cell
Four-dimensional analog of the icosahedron
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system"
600-cell
Type of Dirichlet series associated to number field extensions
algebraically speaking, the case when ρ is a real representation or quaternionic representation. The Artin root number is the subject of significant
Artin_L-function
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
Concept in mathematics
indicates whether a given irreducible character is real, complex, or quaternionic. They are examples of Schur functors. They are defined as follows. Let
Tensor product of representations
Tensor_product_of_representations
U(N) to U(N – 1) states that Example. The unitary symplectic group or quaternionic unitary group, denoted Sp(N) or U(N, H), is the group of all transformations
Restricted_representation
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
Mathematics term
≥ 2. For n ≥ 2, the noncompact Lie group Sp(n, 1) of isometries of a quaternionic hermitian form of signature (n,1) is a simple Lie group of real rank
Kazhdan's_property_(T)
Type of group representation for locally compact groups
Blattner's conjecture Holomorphic discrete series representation Quaternionic discrete series representation Atiyah, Michael; Schmid, Wilfried (1977), "A geometric
Discrete series representation
Discrete_series_representation
Representation of semisimple Lie groups
characters of holomorphic discrete series representations. Quaternionic discrete series representation Bargmann, V (1947), "Irreducible unitary representations
Holomorphic discrete series representation
Holomorphic_discrete_series_representation
Differential geometry concept
Wolf space to each of the simple complex Lie groups. Quaternionic discrete series representation Besse, Arthur L. (2008), Einstein Manifolds, Classics
Quaternion-Kähler symmetric space
Quaternion-Kähler_symmetric_space
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
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
group is the metaplectic group, which appears in infinite-dimensional representation theory and physics. When one takes for K ⊂ π 1 ( G ) {\displaystyle
Simple_Lie_group
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
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system"
Snub_24-cell
Matrix equal to its conjugate-transpose
Hermitian matrices are utilized in tasks like Fourier analysis and signal representation. The eigenvalues and eigenvectors of Hermitian matrices play a crucial
Hermitian_matrix
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8 {\displaystyle
Dual_snub_24-cell
Special mathematical functions defined on the surface of a sphere
certain spin representations of SO(3), with respect to the action by quaternionic multiplication. Spherical harmonics can be separated into two sets of
Spherical_harmonics
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
Natural number
e_{i}\pm e_{j}:1\leq i<j\leq 4\}} in four-dimensional Euclidean space. In quaternionic form, the same configuration may be identified with the 24 unit Hurwitz
24_(number)
Random matrix with gaussian entries
{\displaystyle M^{*}} is its transpose. If M {\displaystyle M} is complex or quaternionic, then M ∗ {\displaystyle M^{*}} is its conjugate transpose. λ 1 , …
Gaussian_ensemble
Mathematical operation
transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a
Cayley_transform
Regular object in four dimensional geometry
16 ] R q 7 , q 8 {\displaystyle [16]R_{q7,q8}} is the conventional representation for all [16] congruent plane displacements. These rotation classes are
24-cell
Correspondence between quaternions and 3D rotations
{\displaystyle {\vec {u}}} that specifies a rotation as to axial vectors. In quaternionic formalism the choice of an orientation of the space corresponds to order
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
geometry used to describe the physical phenomena of quantum physics Quaternionic analysis Ramsey theory the study of the conditions in which order must
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
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
(2017), "The Howe duality conjecture: quaternionic case", in Cogdell, J.; Kim, J.-L.; Zhu, C.-B. (eds.), Representation Theory, Number Theory, and Invariant
Theta_correspondence
Sporadic simple group
McL is the only sporadic group to admit irreducible representations of quaternionic type. It has 2 such representations, one of dimension 3520 and one of
McLaughlin_sporadic_group
(pseudo-)Riemannian manifold whose geodesics are reversible
quaternion-Kähler if and only if isotropy representation of K contains an Sp(1) summand acting like the unit quaternions on a quaternionic vector space. Thus the quaternion-Kähler
Symmetric_space
Smooth manifold with an inner product on each tangent space
metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley plane are analogues of the real projective
Riemannian_manifold
Algebraic structure designed for geometry
analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis
Geometric_algebra
Representation theory
the Weyl group of A. The group G = SL(2,C) acts transitively on the quaternionic upper half space H 3 = { x + y i + t j ∣ t > 0 } {\displaystyle {\mathfrak
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Super vector space forming base superspace for supersymmetric field theories
becomes the real dimension. On the other hand if the reality structure is quaternionic or complex (hermitian), the real dimension is double the complex dimension
Super_Minkowski_space
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
function whose domain is the entire complex plane. Quaternionic function: a function whose domain is quaternionic. Hypercomplex function: a function whose domain
List_of_types_of_functions
Geometric concept of a 2D space with "points at infinity" adjoined
pappian planes) serve as fundamental examples in algebraic geometry. The quaternionic projective plane HP2 is also of independent interest. By Wedderburn's
Projective_plane
Finite simple group type not classified as Lie, cyclic or alternating
a type 2-3-3 triangle J2 is the group of automorphisms preserving a quaternionic structure (modulo its center). Consists of subgroups which are closely
Sporadic_group
Concept in differential geometry
Date incompatibility (help) Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc., 71, 3, 1 (3): 526–7, doi:10
Holonomy
Method of constructing instanton solutions
Let x be the 4-dimensional Euclidean spacetime coordinates written in quaternionic notation x i j = ( z 2 z 1 − z 1 ¯ z 2 ¯ ) . {\displaystyle
ADHM_construction
Theorem in quantum mechanics
measurements are defined must be a real or complex Hilbert space, or a quaternionic module. (Gleason's argument is inapplicable if, for example, one tries
Gleason's_theorem
Lie groups and their associated Lie algebras
Lie group#Full classification Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics
Table_of_Lie_groups
Fiber bundle whose fibers are group torsors
S^{4n+3}} is a principal S p ( 1 ) {\displaystyle Sp(1)} -bundle over quaternionic projective space H P n {\displaystyle \mathbb {H} \mathbb {P} ^{n}}
Principal_bundle
Matrix representing a Euclidean rotation
\mathrm {SO} (3).} For a detailed account of the SU(2)-covering and the quaternionic covering, see spin group SO(3). Many features of these cases are the
Rotation_matrix
operators on an infinite-dimensional real, complex or quaternionic Hilbert space. The quaternionic space is defined as all sequences x = (xi) with xi in
Jordan_operator_algebra
Equations describing classical electromagnetism
geometric algebra formulation and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation was used. Maxwell's equations
Maxwell's_equations
Classification in abstract algebra
subalgebra (n odd), determining whether the central simple factor is split or quaternionic. Each of these properties depends only on the signature p − q modulo
Classification of Clifford algebras
Classification_of_Clifford_algebras
Four finite groups derived from the Leech lattice
Hall–Janko group J2 (order 604,800) as the quotient of the group of quaternionic automorphisms of Λ by the group ±1 of scalars. The seven simple groups
Conway_group
Four-dimensional associative algebra over the reals
2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature, k is replaced
Split-quaternion
Fringe theory of physics
single Lie group geometry—specifically, excitations of the noncompact quaternionic real form of the largest simple exceptional Lie group, E8. A Lie group
An Exceptionally Simple Theory of Everything
An_Exceptionally_Simple_Theory_of_Everything
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
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
Every polynomial has a real or complex root
Eilenberg–Niven theorem, a generalization of the theorem to polynomials with quaternionic coefficients and variables Hilbert's Nullstellensatz, a generalization
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Spin representations of the SO(3) group
constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that
Spinors_in_three_dimensions
researcher Katrin Leschke (born 1968), German differential geometer, quaternionic analyst, and minimal surface theorist Nandi Olive Leslie, American industrial
List_of_women_in_mathematics
Geometric model of the physical space
5. ISBN 978-0-19-960139-4. Morais, João Pedro; et al. (2014). Real Quaternionic Calculus Handbook. Springer Science & Business Media. pp. 1–13. ISBN 978-3-0348-0622-0
Three-dimensional_space
Structure group sub-bundle on a tangent frame bundle
a reduction of the frame bundle, then the solder form consists of a representation ρ of G on Rn and an isomorphism of bundles θ : TM → Q ×ρ Rn. Several
G-structure_on_a_manifold
Study of complex manifolds and several complex variables
complex structures I , J , K {\displaystyle I,J,K} which satisfy the quaternionic relations I 2 = J 2 = K 2 = I J K = − Id {\displaystyle
Complex_geometry
Completion of the usual space with "points at infinity"
naturally to the case where K is a division ring; see, for example, Quaternionic projective space. The notation PG(n, K) is sometimes used for Pn(K).
Projective_space
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
Group of unitary matrices
Classical Mechanics (Second ed.). Springer. p. 225. Baez, John. "Symplectic, Quaternionic, Fermionic". Retrieved 1 February 2012. Grove (2002), Theorem 10.3. Grove
Unitary_group
Generalized sphere of dimension n (mathematics)
-sphere, Lie group structure Sp(1) = SU(2). 4-sphere Homeomorphic to the quaternionic projective line, H P 1 {\displaystyle \mathbf {HP} ^{1}} . SO
N-sphere
In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When n = 4, the Dirac operator is sometimes referred to as
Clifford_analysis
Double cover Lie group of the special orthogonal group
{\text{SO}} (3)\cong \mathbb {RP} ^{3}} (shown using the axis-angle representation). The proof uses known results in algebraic topology. The same argument
Spin_group
generalization of the Pauli matrices; these matrices are one notable representation of the infinitesimal generators of the special unitary group SU(3).
List_of_named_matrices
Special type of principal bundle
four-dimensional sphere S 4 {\displaystyle S^{4}} , which include the quaternionic Hopf fibration, can be used to describe hypothetical magnetic monopoles
Principal_SU(2)-bundle
Complex vector of electromagnetic fields
transition is made: With the advent of spinor calculus that superseded the quaternionic calculus, the transformation properties of the Riemann-Silberstein vector
Riemann–Silberstein_vector
Low-rank isomorphisms in mathematics
)\times \mathrm {SL} (2,\mathbf {R} )\to \mathrm {SO} (2,2).} On the quaternionic real form one recovers the compact case S U ( 2 ) × S U ( 2 ) → S O (
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
Generalization of a polytope in real space
in this 20-gonal projection. Quaternionic polytope Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation for Shephard groups. Mathematische
Complex_polytope
Development of linear transformations forming the Lorentz group
2}\end{aligned}}\end{matrix}}} Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms
History of Lorentz transformations
History_of_Lorentz_transformations
Mathematical concept
Sabadini; M Shapiro; F Sommen (eds.). Hypercomplex analysis (Conference on quaternionic and Clifford analysis; proceedings ed.). Birkhäuser. p. 168. ISBN 978-3-7643-9892-7
Seven-dimensional cross product
Seven-dimensional_cross_product
QUATERNIONIC REPRESENTATION
QUATERNIONIC REPRESENTATION
Biblical
a guard of four soldiers,...and delivered him to four quaternions of soldiers to guard him...
Girl/Female
Hindu, Indian
Representation of Love
QUATERNIONIC REPRESENTATION
QUATERNIONIC REPRESENTATION
Girl/Female
Muslim/Islamic
Beloved sweetheart, darling
Boy/Male
Tamil
Equinox
Girl/Female
Gujarati, Hindu, Indian
Good Planning; Best Creation
Boy/Male
Australian, Finnish
Valley
Girl/Female
Muslim
Blessed, Fortunate
Boy/Male
Gujarati, Hindu, Indian, Punjabi, Sikh
Glow; Light
Girl/Female
Bengali, Gujarati, Hindu, Indian
Viewer
Boy/Male
Bengali, Hindu, Indian, Jain
Happy; One of the Ragas; Ansh of Laxmi
Girl/Female
Latin
Laurel.
Boy/Male
Arabic, Australian
Lion; Kind Heart
QUATERNIONIC REPRESENTATION
QUATERNIONIC REPRESENTATION
QUATERNIONIC REPRESENTATION
QUATERNIONIC REPRESENTATION
QUATERNIONIC REPRESENTATION
n.
A description or statement; as, the representation of an historian, of a witness, or an advocate.
n.
A likeness, a picture, or a model; as, a representation of the human face, or figure, and the like.
n.
A portrait or representation of the face of our Savior on the alleged handkerchief of Saint Veronica, preserved at Rome; hence, a representation of this portrait, or any similar representation of the face of the Savior. Formerly called also Vernacle, and Vernicle.
n.
A vessel similar to that described in the first definition above, or the representation of one in a solid block of stone, or the like, used for an ornament, as on a terrace or in a garden. See Illust. of Niche.
n.
A set of four parts, things, or person; four things taken collectively; a group of four words, phrases, circumstances, facts, or the like.
n.
A dramatic performance; as, a theatrical representation; a representation of Hamlet.
n.
A general form or structure common to a number of individuals; hence, the ideal representation of a species, genus, or other group, combining the essential characteristics; an animal or plant possessing or exemplifying the essential characteristics of a species, genus, or other group. Also, a group or division of animals having a certain typical or characteristic structure of body maintained within the group.
n.
The act or art of expressing by means of types or symbols; emblematical or hieroglyphic representation.
n.
The number four.
n.
The body of those who act as representatives of a community or society; as, the representation of a State in Congress.
n.
The pictorial representation of a scene; a sketch, /ither drawn or painted; as, a fine view of Lake George.
n.
The number four; a collection of four things; a quaternion.
n.
In dramatic composition, one of the principles by which a uniform tenor of story and propriety of representation are preserved; conformity in a composition to these; in oratory, discourse, etc., the due subordination and reference of every part to the development of the leading idea or the eastablishment of the main proposition.
n.
The turning factor of a quaternion.
n.
In the quaternion analysis, a quantity that has magnitude, but not direction; -- distinguished from a vector, which has both magnitude and direction.
n.
A representation of the world.
n.
A word of four syllables; a quadrisyllable.
v. t.
To divide into quaternions, files, or companies.
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.
a.
Implying representation; representative.