|
Pseudoreal
In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space ''V'' with an invariant quaternionic structure, i.e., an antilinear equivariant map :j\colon V\to V which satisfies :j^2=-1. Together with the imaginary unit ''i'' and the antilinear map ''k'' := ''ij'', ''j'' equips ''V'' with the structure of a quaternionic vector space (i.e., ''V'' becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation of a group ''G'' is a group homomorphism ''φ'': ''G'' → GL(''V'', H), the group of invertible quaternion-linear transformations of ''V''. In particular, a quaternionic matrix representation of ''g'' assigns a square matrix of quaternions ''ρ''(g) to each element ''g'' of ''G'' such that ''ρ''(e) is the identity matrix and :\rho(gh)=\rho(g)\rho(h)\textg, h \in G. Quaternionic representations of associative and Lie algebras can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Representation
In the mathematical field of representation theory a real representation is usually a representation on a real vector space ''U'', but it can also mean a representation on a complex vector space ''V'' with an invariant real structure, i.e., an antilinear equivariant map :j\colon V\to V which satisfies :j^2=+1. The two viewpoints are equivalent because if ''U'' is a real vector space acted on by a group ''G'' (say), then ''V'' = ''U''⊗C is a representation on a complex vector space with an antilinear equivariant map given by complex conjugation. Conversely, if ''V'' is such a complex representation, then ''U'' can be recovered as the fixed point set of ''j'' (the eigenspace with eigenvalue 1). In physics, where representations are often viewed concretely in terms of matrices, a real representation is one in which the entries of the matrices representing the group elements are real numbers. These matrices can act either on real or complex column vectors. A real representati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms). It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Representation
In mathematical field of representation theory, a symplectic representation is a representation of a group or a Lie algebra on a symplectic vector space (''V'', ''ω'') which preserves the symplectic form ''ω''. Here ''ω'' is a nondegenerate skew symmetric bilinear form :\omega\colon V\times V \to \mathbb F where F is the field of scalars. A representation of a group ''G'' preserves ''ω'' if :\omega(g\cdot v,g\cdot w)= \omega(v,w) for all ''g'' in ''G'' and ''v'', ''w'' in ''V'', whereas a representation of a Lie algebra g preserves ''ω'' if :\omega(\xi\cdot v,w)+\omega(v,\xi\cdot w)=0 for all ''ξ'' in g and ''v'', ''w'' in ''V''. Thus a representation of ''G'' or g is equivalently a group or Lie algebra homomorphism from ''G'' or g to the symplectic group Sp(''V'',''ω'') or its Lie algebra sp(''V'',''ω'') If ''G'' is a compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Representation
In mathematics, the spin representations are particular projective representations of the orthogonal group, orthogonal or special orthogonal groups in arbitrary dimension and metric signature, signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representation of a Lie group, representations of the spin groups, which are Double covering group, double covers of the special orthogonal groups. They are usually studied over the real number, real or complex numbers, but they can be defined over other field (mathematics), fields. Elements of a spin representation are called spinors. They play an important role in the physics, physical description of fermions such as the electron. The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spinor Group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Lie group, Spin(''n'') therefore shares its dimension, , and its Lie algebra with the special orthogonal group. For , Spin(''n'') is simply connected and so coincides with the universal cover of SO(''n''). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −. Spin(''n'') can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(''n''). A distinct article discusses the spin representations. Motivation and physical interpretation The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. The circle group forms a subgroup of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure \theta: \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional object has an infinite number of possible central axes and rotational directions. If the rotation axis passes internally through the body's own center of mass, then the body is said to be ''autorotating'' or '' spinning'', and the surface intersection of the axis can be called a ''pole''. A rotation around a completely external axis, e.g. the planet Earth around the Sun, is called ''revolving'' or ''orbiting'', typically when it is produced by gravity, and the ends of the rotation axis can be called the ''orbital poles''. Mathematics Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions (in a plane and in space, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Structure
In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map \sigma: \to \,, with \sigma (z)=, giving the "canonical" real structure on \,, that is =\oplus i\,. The conjugation map is antilinear: \sigma (\lambda z)=\sigma(z)\, and \sigma (z_1+z_2)=\sigma(z_1)+\sigma(z_2)\,. Vector space A real structure on a complex vector space ''V'' is an antilinear involution \sigma: V \to V. A real structure defines a real subspace V_ \subset V, its fixed locus, and the natural map : V_ \otimes_ \to V is an isomorphism. Conversely any vector space that is the complexification of a real vector space has a natural real structure. One first notes that every complex space ''V'' has a realification obtained by taking the same vectors as in the original ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate Representation
In mathematics, if is a group and is a representation of it over the complex vector space , then the complex conjugate representation is defined over the complex conjugate vector space as follows: : is the conjugate of for all in . is also a representation, as one may check explicitly. If is a real Lie algebra and is a representation of it over the vector space , then the conjugate representation is defined over the conjugate vector space as follows: : is the conjugate of for all in .This is the mathematicians' convention. Physicists use a different convention where the Lie bracket of two real vectors is an imaginary vector. In the physicist's convention, insert a minus in the definition. is also a representation, as one may check explicitly. If two real Lie algebras have the same complexification, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different. See spinor for some examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
