56 Sentences With "complexification" | Random Sentence Generator

Lynch's 218-episode sequel to his 219 series is a continuation that could better be described as a complexification, not only for its conceptual expansiveness but also for its stretches of intensive narrative languor.

In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the compact Lie group.

Aut EC is the complexification of the compact Lie group Aut E in GL(EC). This follows because the Lie algebras of Aut EC and Aut E consist of derivations of the complex and real Jordan algebras EC and E. Under the isomorphism identifying End EC with the complexification of End E, the complex derivations is identified with the complexification of the real derivations.

In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.

The universal property implies that the universal complexification is unique up to complex analytic isomorphism.

The complexification of a simple Euclidean Jordan algebra is a simple complex Jordan algebra which is also separable, i.e. its trace form is non-degenerate. Conversely, using the existence of a real form of the Lie algebra of the structure group, it can be shown that every complex separable simple Jordan algebra is the complexification of a simple Euclidean Jordan algebra. To verify that the complexification of a simple Euclidean Jordan algebra E has no ideals, note that if F is an ideal in EC then so too is F⊥, the orthogonal complement for the trace norm.

There are other closed subgroups of the complexification of a compact connected Lie group G which have the same complexified Lie algebra. These are the other real forms of GC.

The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.) Over the complex numbers the semisimple Lie algebras are classified by their Dynkin diagrams, of types "ABCDEFG". If L is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless L is already the complexification of a Lie algebra, in which case the complexification of L is a product of two copies of L. This reduces the problem of classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e.

In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.

For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite- dimensional representations of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup of the complex general linear group. It consists of operators with polar decomposition , where is a unitary operator in the compact group and is a skew-adjoint operator in its Lie algebra.

Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group K with complexification G, the inclusion from K into the complex reductive group G(C) is a homotopy equivalence, with respect to the classical topology on G(C). For example, the inclusion from the unitary group U(n) to GL(n,C) is a homotopy equivalence.

13 – Arnold, Vladimir, 1997, Toronto Lectures, Lecture 2: Symplectization, Complexification and Mathematical Trinities, June 1997 (last updated August, 1998). TeX, PostScript, PDF and can be related to many further objects, including E7 and E8, as discussed at trinities.

A real circle, defined by its center point (x0,y0) and radius r (all three of which are real numbers) may be described as the set of real solutions to the equation :(x-x_0)^2+(y-y_0)^2=r^2. Converting this into a homogeneous equation and taking the set of all complex-number solutions gives the complexification of the circle. The two circular points have their name because they lie on the complexification of every real circle. More generally, both points satisfy the homogeneous equations of the type :Ax^2 + Ay^2 + 2B_1xz + 2B_2yz - Cz^2 = 0.

In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space. For example, when with the standard complex conjugation :\chi(z_1,\ldots,z_n) = (\bar z_1,\ldots,\bar z_n) the invariant subspace is just the real subspace .

Using complexification, rewrote the formula (1) into : where CS(S^3\backslash K) is called the Chern–Simons invariant. They showed that there is a clear relation between the complexified colored Jones polynomial and Chern–Simons theory from a mathematical point of view.

They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.

Using the Lie correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups. In the case of linear algebraic groups, the notions of complexification and real form have a natural description in the language of algebraic geometry.

In his work Chaosmosis, Guattari explains that "rather than moving in the direction of reductionist modifications which simplify the complex", schizoanalysis "will work towards its complexification, its processual enrichment, towards the consistency of its virtual lines of bifurcation and differentiation, in short towards its ontological heterogeneity".Guattari (1992, 61).

Ben-Jacob E: Bacterial self-organization: co-enhancement of complexification and adaptability in a dynamic environment. Phil Trans R Soc Lond A 2003, 361:1283-1312.Ben-Jacob E, Cohen I, Gutnick DL: Cooperative organization of bacterial colonies: from genotype to morphotype. Annu Rev Microbiol 1998, 52:779-806.

Every complex, semisimple Lie algebra has a split real form and a compact real form; the former is called a complexification of the latter two. The Lie algebra of is semisimple and is denoted . Its split real form is and its compact real form is . These correspond to the Lie groups and respectively.

This embeds as a subgroup of , and hence we may realise as a subgroup of . Furthermore, is the complexification of . In the complex case, quadratic forms are determined uniquely up to isomorphism by the dimension of . Concretely, we may assume and :Q(z_1,\ldots, z_n) = z_1^2+ z_2^2+\cdots+z_n^2.

As with the inclusion of points at infinity and complexification of real polynomials, this allows some theorems to be stated more simply without exceptions and for a more regular algebraic analysis of the geometry. Viewed in terms of homogeneous coordinates, a real vector space of homogeneous coordinates of the original geometry is complexified. A point of the original geometric space is defined by an equivalence class of homogeneous vectors of the form , where is an nonzero complex value and is a real vector. A point of this form (and hence belongs to the original real space) is called a real point, whereas a point that has been added through the complexification and thus does not have this form is called an imaginary point.

Other structures considered on include the one of a pseudo-Euclidean space, symplectic structure (even ), and contact structure (odd ). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates. is also a real vector subspace of which is invariant to complex conjugation; see also complexification.

On page 665 of Elements of Quaternions Hamilton defines a biquaternion to be a quaternion with complex number coefficients. The scalar part of a biquaternion is then a complex number called a biscalar. The vector part of a biquaternion is a bivector consisting of three complex components. The biquaternions are then the complexification of the original (real) quaternions.

In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include their scaling ("multiplication") by complex numbers. Any basis for (a space over the real numbers) may also serve as a basis for over the complex numbers.

If is a Lie group, a universal complexification is given by a complex Lie group and a continuous homomorphism with the universal property that, if is an arbitrary continuous homomorphism into a complex Lie group , then there is a unique complex analytic homomorphism such that . Universal complexifications always exist and are unique up to a unique complex analytic isomorphism (preserving inclusion of the original group).

Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra (the Iwasawa decomposition), and this provides a canonical bicrossproduct quantum group associated to g. For su(2) one obtains a quantum group deformation of the Euclidean group E(3) of motions in 3 dimensions.

Elements of a spin representation are called spinors. They play an important role in the 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 complexification of the vector representation.

Vyacheslav Ivanov and Vladimir Toporov studied the origin of ancient Slavic themes in the common substratum represented by Proto-Indo-European religion and what Georges Dumézil studied as the "trifunctional hypothesis". Marija Gimbutas, instead, found Slavic religion to be a clear result of the overlap of Indo-European patriarchism and pre-Indo-European matrifocal beliefs. Boris Rybakov emphasised the continuity and complexification of Slavic religion through the centuries.

There is thus a group homomorphism whose kernel has two elements denoted , where is the identity element. Thus, the group elements and of are equivalent after the homomorphism to ; that is, for any in . The groups and are all Lie groups, and for fixed they have the same Lie algebra, . If is real, then is a real vector subspace of its complexification , and the quadratic form extends naturally to a quadratic form on .

For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant convex cone. When one is properly contained in the other, there is a continuum of intermediate invariant convex cones. Invariant convex cones arise in the analysis of holomorphic semigroups in the complexification of the Lie group, first studied by Grigori Olshanskii. They are naturally associated with Hermitian symmetric spaces and their associated holomorphic discrete series.

He summarized this work in terms of the complexification of the two-form fiber over space-time.Carl Brans Complex Structures and the Einstein Equations J. Math. Phys. 15 1559 (1974). He also worked on certain questions related to the apparently circular argument in proofs of Bell's theorem in which the hidden variables are a priori assumed to not influence detector settings,Carl Brans Bell's Theorem does not eliminate fully causal Hidden Variables Int.

The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.

This implies in particular that is compact. There is a more direct proof of compactness using symmetry groups. Given a Jordan frame (ei) in E, for every a in A there is a k in U = Γu(A) such that with (and if a is invertible). In fact, if (a,b) is in X then it is equivalent to k(c,d) with c and d in the unital Jordan subalgebra , which is the complexification of .

If the original group is linear, so too is the universal complexification and the homomorphism between the two is an inclusion. give an example of a connected real Lie group for which the homomorphism is not injective even at the Lie algebra level: they take the product of by the universal covering group of and quotient out by the discrete cyclic subgroup generated by an irrational rotation in the first factor and a generator of the center in the second.

In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.

Given a field extension, one can "extend scalars" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via complexification. In addition to vector spaces, one can perform extension of scalars for associative algebras defined over the field, such as polynomials or group algebras and the associated group representations. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally.

The original (K-L) case is then about the details of decomposing :B\backslash G/ B, a classical theme of the Bruhat decomposition, and before that of Schubert cells in a Grassmannian. The L-V case takes a real form of G, a maximal compact subgroup in that semisimple group , and makes the complexification K of . Then the relevant object of study is :K\backslash G/ B. In March 2007, it was announced that the L–V polynomials had been calculated for the split form of E8.

In the mid-1990s, McRobbie describes the occurrence of a "complexification of backlash" towards feminism, marking a decisive shift where the forces opposing gender equality and the visibility of women in positions of power blamed feminism for the rise in divorce rates, crises in masculinity and the "feminisation of the curriculum in schools". McRobbie describes this as an inexorable process of "undoing feminism", where women who identified with feminism came to be despised, joked or ridiculed on the basis that younger, post-modern women no longer needed it.

Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C).

The complexification of a real vector space results in a complex vector space (over the complex number field). To "complexify" a space means extending ordinary scalar multiplication of vectors by real numbers to scalar multiplication by complex numbers. For complexified inner product spaces, the complex inner product on vectors replaces the ordinary real-valued inner product, an example of the latter being the dot product. In mathematical physics, when we complexify a real coordinate space Rn we create a complex coordinate space Cn, referred to in differential geometry as a "complex manifold".

In particular gave a general method for deducing properties of the spherical transform for a real semisimple group from that of its complexification. One of the principal applications and motivations for the spherical transform was Selberg's trace formula. The classical Poisson summation formula combines the Fourier inversion formula on a vector group with summation over a cocompact lattice. In Selberg's analogue of this formula, the vector group is replaced by G/K, the Fourier transform by the spherical transform and the lattice by a cocompact (or cofinite) discrete subgroup.

Marija Gimbutas, instead, found Slavic religion to be a clear result of the overlap of Indo-European patriarchism and pre-Indo-European matrifocal beliefs. Boris Rybakov emphasised the continuity and complexification of Slavic religion through the centuries. Laruelle observed that Rodnovery is in principle a decentralised movement, with hundreds of groups coexisting without submission to a central authority. Therefore, socio-political views can vary greatly from one group to another, from one adherent to another, ranging from extreme pacifism to militarism, from apoliticism and anarchism to left-wing and to right-wing positions.

Therefore, these living things did not necessarily evolve through a gradual process of natural selection. Rather, he posited, the process of evolution experiences jumps in complexity (such as the emergence of a self-reflective universe, or noosphere), in a sort of qualitative punctuated equilibrium. Finally, the complexification of human cultures, particularly language, facilitated a quickening of evolution in which cultural evolution occurs more rapidly than biological evolution. Recent understanding of human ecosystems and of human impact on the biosphere have led to a link between the notion of sustainability with the "co-evolution" and harmonization of cultural and biological evolution.

The semigroup is made up of those elements in the complexification which, when acting on the Hermitian symmetric space of compact type, leave invariant the bounded domain corresponding to the noncompact dual. The semigroup acts by contraction operators on the holomorphic discrete series; its interior acts by Hilbert–Schmidt operators. The unitary part of their polar decomposition is the operator corresponding to an element in the original real Lie group, while the positive part is the exponential of an imaginary multiple of the infinitesimal operator corresponding to an element in the maximal cone. A similar decomposition already occurs in the semigroup.

In finite dimensions the existence of a fixed point can often be deduced from the Brouwer fixed point theorem without any appeal to holomorphicity of the mapping. In the case of bounded symmetric domains with the Bergman metric, and showed that the same scheme of proof as that used in the Earle-Hamilton theorem applies. The bounded symmetric domain D = G / K is a complete metric space for the Bergman metric. The open semigroup of the complexification Gc taking the closure of D into D acts by contraction mappings, so again the Banach fixed-point theorem can be applied.

In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations of the ring C[z,z−1]. There are some related Lie algebras defined over finite fields, that are also called Witt algebras. The complex Witt algebra was first defined by Cartan (1909), and its analogues over finite fields were studied by Witt in the 1930s.

No longer was the artist restricted to a principle (or set of principles). The liberation from academicism offered by Cubism, as instigated by Cézanne, resulted from the fact that the artist was no longer restricted to the representation of the subject (or the world) as seen in a photograph. No longer restricted to the imitative description of nature, and despite the complexification of visual stimuli (many views instead of one), the technique of painting became simple and direct. In 1912 the photographic motion studies of Eadweard Muybridge and Étienne-Jules Marey particularly interested artists of the Section d'Or, including Jean Metzinger, Marcel Duchamp and Albert Gleizes.

The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrically charged fermions, most notably the electron. Strictly speaking, the spin group describes a fermion in a zero-dimensional space; but of course, space is not zero-dimensional, and so the spin group is used to define spin structures on (pseudo-)Riemannian manifolds: the spin group is the structure group of a spinor bundle. The affine connection on a spinor bundle is the spin connection; the spin connection is useful as it can simplify and bring elegance to many intricate calculations in general relativity.

He states that it is a notion that can interpret the enormous expansion and complexification of the physical universe (from the Big Bang outward), as well as the evolution of life here on earth and the gradual emergence of human historical existence. The whole vast process manifests (in varying degrees) serendipitous creativity, an everflowing coming into being of new modes of reality.Gordon D. Kaufman – God, mystery, diversity: Christian theology in a pluralistic world, Fortress Press, 1996, page 101–102, retrieved 3-17-09 In his book, In the beginning—creativity, he says this creative process is God. Creativity, as metaphor, and as defined in the concept of evolution, has possibilities for constructing a new concept of God.

Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact group SO(3) or SU(2) as gauge group, as opposed to the non-compact groups SO(3,1) or SL(2.C). The compactness of the Lie group avoids some thus- far unsolved difficulties in the quantization of gauge theories of noncompact lie groups, and is responsible for the discreteness of the area and volume spectra. The theory involving the Immirzi parameter is necessary to resolve an ambiguity in the process of complexification.

Alexander von Humboldt in 1843 Astronomer Carl Sagan While the emerging field of Big History in its present state is generally seen as having emerged in the past two decades beginning around 1990, there have been numerous precedents going back almost 150 years. In the mid-19th century, Alexander von Humboldt's book Cosmos, and Robert Chambers' 1844 book Vestiges of the Natural History of Creation were seen as early precursors to the field. In a sense, Darwin's theory of evolution was, in itself, an attempt to explain a biological phenomenon by examining longer term cause-and-effect processes. In the first half of the 20th century, secular biologist Julian Huxley originated the term "evolutionary humanism", while around the same time the French Jesuit paleontologist Pierre Teilhard de Chardin examined links between cosmic evolution and a tendency towards complexification (including human consciousness), while envisaging compatibility between cosmology, evolution, and theology.

He has written fourteen popular books on science. These include Paradigms Lost: Images of Man in the Mirror of Science (Morrow, NY, 1989), which addresses several of the most puzzling controversies in modern science, Searching for Certainty: What Scientists Can Know About the Future (Morrow, NY, 1991), a volume dealing with problems of scientific prediction and explanation of everyday events like the weather, stock market price movements and the outbreak of warfare, and Complexification (HarperCollins, NY, 1994), a study of complex systems and the manner in which they give rise to counterintuitive, surprising behavior. Dr. Casti has also written three popular volumes on mathematics: Five Golden Rules: Great Theories of 20th- Century Mathematics---and Why They Matter; a sequel, Five More Golden Rules (1995, 2000) both published by John Wiley & Sons (New York); and Mathematical Mountaintops: The Five Most Famous Problems of All Time, published and later recalled by Oxford University Press (New York).

If the reflection group W corresponds to the Weyl group of a compact semisimple group K with maximal torus T, then the Kostant polynomials describe the structure of the de Rham cohomology of the generalized flag manifold K/T, also isomorphic to G/B where G is the complexification of K and B is the corresponding Borel subgroup. Armand Borel showed that its cohomology ring is isomorphic to the quotient of the ring of polynomials by the ideal generated by the invariant homogeneous polynomials of positive degree. This ring had already been considered by Claude Chevalley in establishing the foundations of the cohomology of compact Lie groups and their homogeneous spaces with André Weil, Jean-Louis Koszul and Henri Cartan; the existence of such a basis was used by Chevalley to prove that the ring of invariants was itself a polynomial ring. A detailed account of Kostant polynomials was given by and independently as a tool to understand the Schubert calculus of the flag manifold.

There is a notable technological complexification coinciding with the replacement of Neanderthals with EEMH in the archaeological record, and so the terms "Middle Palaeolithic" and "Upper Palaeolithic" were created to distinguish between these two time periods. Largely based on Western European archaeology, the transition was dubbed the "Upper Palaeolithic Revolution," (extended to be a worldwide phenomenon) and the idea of "behavioural modernity" became associated with this event and early modern cultures. It is largely agreed that the Upper Palaeolithic seems to feature a higher rate of technological and cultural evolution than the Middle Palaeolithic, but it is debated if behavioural modernity was truly an abrupt development or was a slow progression initiating far earlier than the Upper Paleolithic, especially when considering the non-European archaeological record. Behaviourly modern practices include: the production of microliths, the common use of bone and antler, the common use of grinding and pounding tools, high quality evidence of body decoration and figurine production, long distance trade networks, and improved hunting technology.

A Lie group is a group that is also a smooth manifold. Many classical groups of matrices over the real or complex numbers are Lie groups.. Many of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields. The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply. This theory can be extended to finite-dimensional representations of semisimple Lie groups using Weyl's unitary trick: each semisimple real Lie group G has a complexification, which is a complex Lie group Gc, and this complex Lie group has a maximal compact subgroup K. The finite-dimensional representations of G closely correspond to those of K. A general Lie group is a semidirect product of a solvable Lie group and a semisimple Lie group (the Levi decomposition).. The classification of representations of solvable Lie groups is intractable in general, but often easy in practical cases.