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Quaternionic manifold – Concept in geometry Quaternionic matrix – Concept in linear algebra Quaternionic polytope – Concept in geometry Quaternionic projective...

In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than...

In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is...

A quaternionic matrix is a matrix whose elements are quaternions. The quaternions form a noncommutative ring, and therefore addition and multiplication...

representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear...

In geometry, a quaternionic polytope is a generalization of a polytope in real space to an analogous structure in a quaternionic module, where each real...

In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates...

conditions. When S {\displaystyle S} is of quaternionic type, the representation carries an invariant quaternionic structure but no invariant real structure...

theories, (real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively....

In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of...

traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups...

\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...

split 52 4 C3 × A1 Order 2 1 F I 28 Quaternionic projective planes in Cayley projective plane. Hyperbolic quaternionic projective planes in hyperbolic Cayley...

In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital...

In noncommutative algebra, a branch of mathematics, a quaternionic vector space is a module over the quaternions. Since the quaternion algebra is division...

mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field. A quaternionic structure...

structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex. Every hyperkähler manifold is also hypercomplex...

\mathbb {CP} ^{n}} with circles as fibers, and there are also real, quaternionic, and octonionic versions of these fibrations. In particular, the Hopf...

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}...

In four-dimensional geometry, the 24-cell is a convex regular 4-polytope, a four-dimensional analogue of a Platonic solid. It is named for the 24 octahedra...

transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a...

matrices, and the circular symplectic ensemble (CSE) on self dual unitary quaternionic matrices. The distribution of the unitary circular ensemble CUE(n) is...

quaternion (in this case, the sub-field of hypercomplex analysis is called quaternionic analysis). A second instance involves functions of a motor variable where...

orthonormal k-frames in C n {\displaystyle \mathbb {C} ^{n}} and the quaternionic Stiefel manifold V k ( H n ) {\displaystyle V_{k}(\mathbb {H} ^{n})}...

mathematics, a quaternionic discrete series representation is a discrete series representation of a semisimple Lie group G associated with a quaternionic structure...

pseudoreal representation V is necessarily a quaternionic representation: it admits an invariant quaternionic structure, i.e., an antilinear equivariant...

mathematics, a hypertoric variety or toric hyperkähler variety is a quaternionic analog of a toric variety constructed by applying the hyper-Kähler quotient...

Moore, E. H. (1922), "On the determinant of an hermitian matrix with quaternionic elements. Definition and elementary properties with applications." (PDF)...

and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation was used. Maxwell's equations are partial differential equations...

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...

metrics, along with hyperbolic space. The complex projective space, quaternionic projective space, and Cayley plane are analogues of the real projective...

concerning smooth vector bundles on a smooth manifold (real, complex, or quaternionic). His topological variant is about continuous (real or complex) vector...

basis with signature (− − − −) and is given in terms of the following 7 quaternionic triples (omitting the scalar identity element): ( I , j , k ) , ( i ...

Society, 2017) Quaternionic approximation: With application to slice regular functions (with Gal, Birkhäuser/Springer, 2019) Quaternionic de Branges spaces...

{\displaystyle \mathbb {H} ^{\otimes 3}=M(4,\mathbb {H} )} yields a quaternionic matrix and its even subalgebra H ⊗ 2 ⊗ R C {\displaystyle \mathbb {H}...

associate a unique Wolf space to each of the simple complex Lie groups. Quaternionic discrete series representation Besse, Arthur L. (2008), Einstein Manifolds...

n-torus, Tn Real projective space, RPn Complex projective space, CPn Quaternionic projective space, HPn Flag manifold Grassmann manifold Stiefel manifold...

Washington (September 1989). "Nonlinearly Equivalent Representations of Quaternionic 2-Groups" (PDF). Transactions of the American Mathematical Society. 315...

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...

derivative corresponds to the integral, whence the term differintegral. In quaternionic analysis, derivatives can be defined in a similar way to real and complex...

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...

deciding whether a real irreducible representation of G is real, complex or quaternionic, in a specific sense defined below. Much of the content below discusses...

Sp(1)<GL(4n,\mathbf {R} )} G L ( n , H ) {\displaystyle GL(n,\mathbf {H} )} : quaternionic general linear group acting on H n ≅ R 4 n {\displaystyle \mathbf {H}...

representation on a complex vector space that is neither real nor pseudoreal (quaternionic). In other words, the group elements are expressed as complex matrices...

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...

{\displaystyle M^{*}} is its transpose. If M {\displaystyle M} is complex or quaternionic, then M ∗ {\displaystyle M^{*}} is its conjugate transpose. λ 1 , … ...

Eilenberg–Niven theorem, a generalization of the theorem to polynomials with quaternionic coefficients and variables Hilbert's Nullstellensatz, a generalization...

Following a substantial debate, the mainstream shifted from Hamilton's quaternionic system to Gibbs's three-vectors system. This transition led to the prevalent...

doi:10.1016/j.crma.2006.10.019 Kraines, Vivian Yoh (1966), "Topology of quaternionic manifolds", Trans. Am. Math. Soc., 122 (2): 357–367, doi:10...

 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...

geometry used to describe the physical phenomena of quantum physics Quaternionic analysis Ramsey theory the study of the conditions in which order must...

22 579–86 & 24:783–4 1912: Quaternionic form of relativity, Phil. Mag. 14 1912 790–809 1913: Second memoir on quaternionic relativity, Phil. Mag. 15 1913...

pappian planes) serve as fundamental examples in algebraic geometry. The quaternionic projective plane HP2 is also of independent interest. By Wedderburn's...

analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis...

In spin geometry, a spinh structure (or quaternionic spin structure) is a generalization of a spin structure. In mathematics, these are used to describe...

H ) {\displaystyle M(2,\mathbb {H} )} (the two-by-two matrices with quaternionic coefficients) Sp(2), corresponding to B 2 ≅ C 2 {\displaystyle B_{2}\cong...

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...

Zbl 0553.32008. Galicki, K.; Lawson, H. Blaine Jr. (1988). "Quaternionic reduction and quaternionic orbifolds". Mathematische Annalen. 282 (1): 1–21. doi:10...

performed repeatedly. In mathematical physics, the trick illustrates the quaternionic mathematics behind the spin of spinors. As with the plate trick, these...

three families of rank one symmetric spaces, together with real and quaternionic hyperbolic spaces, classification to which must be added one exceptional...

diffeomorphic to the sphere, or isometric to the complex projective space, the quaternionic projective space, or else the Cayley plane F4/Spin(9); see (Brendle &...

{\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...

four-dimensional sphere S 4 {\displaystyle S^{4}} , which include the quaternionic Hopf fibration, can be used to describe hypothetical magnetic monopoles...

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...

function whose domain is the entire complex plane. Quaternionic function: a function whose domain is quaternionic. Hypercomplex function: a function whose domain...

structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler. Hopf, Heinz (1948), "Zur Topologie...

\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...

Couplings in N=2 Supergravity that in this case, the Higgs branch must be a quaternionic Kähler manifold. In extended supergravities with N>2 the moduli space...

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...

single Lie group geometry—specifically, excitations of the noncompact quaternionic real form of the largest simple exceptional Lie group, E8. A Lie group...

show that any complex symplectic representation is a quaternionic representation. Quaternionic representations of finite or compact groups are often...

2006) Manifolds with para-quaternionic structures are studied in differential geometry and string theory. In the para-quaternionic literature, k is replaced...

{1}{Z_{{\text{GSE}}(n)}}}e^{-n\mathrm {tr} H^{2}}} on the space of n × n Hermitian quaternionic matrices, e.g. symmetric square matrices composed of quaternions, H =...

Menger sponge Newton fractal Nova fractal - derived from Newton fractal Quaternionic fractal - three dimensional complex quadratic map Sierpinski carpet Sierpinski...

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...

Date incompatibility (help) Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc., 71, 3, 1 (3): 526–7, doi:10...

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}...

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...

squashed 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...

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...

a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius...

giving the signature see Hirzebruch signature theorem. There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure. Chern–Simons...

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}} ...

manifold. Complex projective space CPn is a 2n-dimensional manifold. Quaternionic projective space HPn is a 4n-dimensional manifold. Manifolds related...

)+s\ \mathbf {v} +t\ \mathbf {w} +\mathbf {v} \times \mathbf {w} .} The quaternionic conjugate of q is defined by q ¯ = t − v , {\displaystyle {\overline...

)\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 (...

is a theorem that generalizes the fundamental theorem of algebra to quaternionic polynomials; that is, polynomials with quaternion coefficients and variables...

first Chern class vanishes. Complex dimension Complex analytic variety Quaternionic manifold Real-complex manifold One must use the open unit ball in the...

of K contains an Sp(1) summand acting like the unit quaternions on a quaternionic vector space. Thus the quaternion-Kähler symmetric spaces are easily...

Routines, World Scientific Publishing Co., 2012, ISBN 978-981-4425-03-2 Quaternionic Quantum Mechanics and Quantum Fields, International Series of Monographs...

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...

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...

Mathematica, v. 141 (2005), pp. 1504–1530. arXiv:math/0402283 Complex forms of quaternionic symmetric spaces, in Complex, contact and symmetric manifolds, Progress...

Generalization of a derivative of a function between two Banach spaces Quaternionic analysis – Function theory with quaternion variable Semi-differentiability –...

geometry and quantum physics). In 1998 he was an Invited Speaker with talk Quaternionic analysis of Riemann surfaces and differential geometry at the International...

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)...

vanishing pure spinor then M is a generalized Calabi–Yau manifold. Almost quaternionic manifold – Concept in geometryPages displaying short descriptions of...

1007/978-1-4612-5648-9_28. ISBN 978-1-4612-5648-9. ————; Weiss, Asia Ivić (July 1999). "Quaternionic modular groups". Linear Algebra and Its Applications. 295 (1): 159–189...

it was proved by Wee Teck Gan and Shuichiro Takeda. The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun. Reductive dual...

station, England Higgs prime, H p n {\displaystyle Hp_{n}} HPN (gene) Quaternionic projective space, H P n {\displaystyle \mathbb {H} \mathrm {P} ^{n}}...

that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so...

certain spin representations of SO(3), with respect to the action by quaternionic multiplication. Spherical harmonics can be separated into two sets of...

theory and quaternion analysis proposing a definition of ‘regular’ for quaternionic functions similar to the definition of holomorphic function by means...

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...

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...

binary icosahedral group. A hyperbolic versor is a generalization of quaternionic versors to indefinite orthogonal groups, such as Lorentz group. It is...

obtained using the Universal coefficient theorem. Complex projective space Quaternionic projective space Lens space Real projective plane See the table of Don...

In spin geometry, a spinh group (or quaternionic spin group) is a Lie group obtained by the spin group through twisting with the first symplectic group...

MR 0290245. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011), "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8...

triangular faces and 640 tetrahedral cells, seen in this 20-gonal projection. Quaternionic polytope Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation...

Classical Mechanics (Second ed.). Springer. p. 225. Baez, John. "Symplectic, Quaternionic, Fermionic". Retrieved 1 February 2012. Grove (2002), Theorem 10.3. Grove...

Schoen's methods is the fact that lattices in the isometry group of the quaternionic hyperbolic space are arithmetic.[GS92] In 1978, Gromov introduced the...

Oppio, Marco (2018). "The correct formulation of Gleason's theorem in quaternionic Hilbert spaces". Annales Henri Poincaré. 19 (11): 3321–3355. arXiv:1803...

Sci. Paris. 260: 5445–5448. Kraines, Vivian Yoh (1965). "Topology of quaternionic manifolds". Bull. Amer. Math. Soc. 71, 3, 1 (3): 526–527. doi:10...

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)....

{\displaystyle \mathbb {CP} ^{2}} ( n = 4 {\displaystyle n=4} ), of the quaternionic projective plane H P 2 {\displaystyle \mathbb {HP} ^{2}} ( n = 8 {\displaystyle...

≥ 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...

attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on H P 2 {\displaystyle \mathbb {HP} ^{2}}...

Commutator subgroup, p. 124–126 Johnson, Norman W.; Weiss, Asia Ivić (1999). "Quaternionic modular groups". Linear Algebra and Its Applications. 295 (1–3): 159–189...

S2CID 119288632. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8...

other physicists of the convenience of the vectorial approach over the quaternionic calculus of William Rowan Hamilton, which was then widely used by British...

and Kottwitz (2005) Harry Reimann, The semi-simple zeta function of quaternionic Shimura varieties, Lecture Notes in Mathematics, 1657, Springer, 1997...

; Pontecorvo, M., eds. (1999), Proceedings of the Second Meeting on Quaternionic Structures in Mathematics and Physics. Dedicated to the Memory of André...

of quantum and classical Fisher information to two-level complex and quaternionic and three-level complex systems". Journal of Mathematical Physics. 37...

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...

{\displaystyle \mathbb {CH} ^{n}} , quaternionic projective spaces H P n {\displaystyle \mathbb {HP} ^{n}} , quaternionic hyperbolic spaces H H n {\displaystyle...

Voight, John (2024). "Rational torsion points on abelian surfaces with quaternionic multiplication". Forum of Mathematics Sigma. 12 e92. doi:10.1017/fms...

{\frac {1}{2}}\right)+i{\mathcal {N}}\left(0,{\frac {1}{2}}\right)} . The quaternionic Ginibre ensemble is defined as G i j ∼ N ( 0 , 1 4 ) + i N ( 0 , 1 4...

} . (Over the 4-sphere S 4 {\displaystyle S^{4}} , this would be the quaternionic Hopf fibration S 7 ↠ S 4 {\displaystyle S^{7}\twoheadrightarrow S^{4}}...

{\displaystyle S^{4}} and hence is a Milnor sphere. Such bundles also include the quaternionic Hopf fibration, whose total space is the ordinary S 7 {\displaystyle...

^{3}-27\delta \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...

S2CID 119288632. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8...

operators on an infinite-dimensional real, complex or quaternionic Hilbert space. The quaternionic space is defined as all sequences x = (xi) with xi in...

quaternion). Versor may also refer to: Hyperbolic versor, a generalization of quaternionic versors Versor (physics), a vector of norm 1 (unit vector) codirectional...

(3): 177 Koca, Mehmet; Al-Ajmi, Mudhahir; Koca, Nazife Ozdes (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8...

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...

described the characters of holomorphic discrete series representations. Quaternionic discrete series representation Bargmann, V (1947), "Irreducible unitary...

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...

"Eichler-Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties" (2018) "Semisimplicity of certain Galois representations...

mathematician specialising in differential geometry and known for her work on quaternionic analysis and Willmore surfaces. She works in England as a reader in 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 ⁡...

H^{2}(X)} (integral cohomology). There is a further, analogous theory with quaternionic (real dimension four) line bundles. This gives rise to one of the Pontryagin...

A:1–9 BHL link via Biodiversity Heritage Library A.W. Conway (1912) "The quaternionic form of relativity", Philosophical Magazine (6) 24:208 George Temple...

the quaternionic projective plane is not its systolically optimal metric, in contrast with the 2-systole in the complex case. While the quaternionic projective...

1)} (groups of matrices with quaternion coefficients which preserve a "quaternionic quadratic form" of signature ( n , 1 ) {\displaystyle (n,1)} ) for n...

ISBN 978-1-56881-220-5. Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E 8...

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...

representation Semisimple Complex representation Real representation Quaternionic representation Pseudo-real representation Symplectic representation Schur's...

algebraically speaking, the case when ρ is a real representation or quaternionic representation. The Artin root number is the subject of significant research...

itself. Analogous Veronese embeddings are constructed for complex and quaternionic projective spaces, as well as for the Cayley plane. Lectures on Discrete...

researcher Katrin Leschke (born 1968), German differential geometer, quaternionic analyst, and minimal surface theorist Nandi Olive Leslie, American industrial...

are polynomials. Positive matrix A matrix with all positive entries. Quaternionic matrix A matrix whose entries are quaternions. Random matrix A matrix...

Cartesian coordinate — the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group...

complex conjugate representations of G . {\displaystyle G.} Definition. A quaternionic representation is a (complex) representation V , {\displaystyle V,} which...

functors. Blattner's conjecture Holomorphic discrete series representation Quaternionic discrete series representation Atiyah, Michael; Schmid, Wilfried (1977)...

{\mathcal {N}}=2} supergravity the relevant scalar manifold must be a quaternionic Kähler manifold. But since these manifolds are not themselves Kähler...

constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that...

transition is made: With the advent of spinor calculus that superseded the quaternionic calculus, the transformation properties of the Riemann-Silberstein vector...

MR 0133834. Floyd, E. E. (1971). "Stiefel-Whitney numbers of quaternionic and related manifolds". Trans. Amer. Math. Soc. 155: 77–94. doi:10...

indicates whether a given irreducible character is real, complex, or quaternionic. They are examples of Schur functors. They are defined as follows. Let...

coordinates, one then defines polar coordinate one-forms on the 4-sphere (the quaternionic projective line) as r d r = + x d x + y d y + z d z + t d t r 2 σ 1 =...

2}\end{aligned}}\end{matrix}}} Arthur W. Conway in February 1911 explicitly formulated quaternionic Lorentz transformations of various electromagnetic quantities in terms...

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:...

1994 Thompson, Robert C. (1997). "The upper numerical range of a quaternionic matrix is not a complex numerical range". Linear Algebra and Its Applications...

concretely, in a real projective space, complex projective space or quaternionic projective space K P n {\displaystyle \mathbb {KP} ^{n}} where K {\displaystyle...