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Split Lie Algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak, \mathfrak) where \mathfrak is a Lie algebra and \mathfrak < \mathfrak is a splitting , where "splitting" means that for all , is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra. Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center. Over an such as the |
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Mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Satake Diagram
In the mathematics, mathematical study of Lie algebras and Lie groups, Satake diagrams are a generalization of Dynkin diagram, Dynkin diagrams that classify involutions of root systems that are relevant in several contexts. They were introduced in and were originally used to classify real form (Lie theory), real simple Lie algebras. Additionally, they also classify symmetric pairs (\mathfrak,\mathfrak) of Lie algebras, where \mathfrak is semisimple. More concretely, given a complex semisimple Lie algebra \mathfrak, the Satake diagrams made from \mathfrak's Dynkin diagram classify the involutions of \mathfrak's root system that extend to an anti-linear involutive automorphism of \mathfrak. The fixed points \mathfrak^\sigma are then a real form of \mathfrak. The same Satake diagrams also classify the involutions of \mathfrak's root system that extend to a (linear) involutive automorphism of \mathfrak. The fixed points \mathfrak form a complex Lie subalgebra of \mathfrak, so that ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Split-complex Number
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+yj . The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number with its conjugate is N(z) := zz^* = x^2 - y^2, an isotropic quadratic form. The collection of all split-complex numbers z=x+yj for forms an algebra over the field of real numbers. Two split-complex numbers and have a product that satisfies N(wz)=N(w)N(z). This composition of over the algebra product makes a composition algebra. A similar algebra based on and component-wise operations of addition and multiplication, where is the quadratic form on also forms a quadratic space. The ring isomorphism \begin D &\to \mathbb^2 \\ x + yj &\mapsto (x - y, x + y) \end is an isometry of quadratic spaces. Split-complex numbers have many other na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Real Form
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0: : \mathfrak\simeq\mathfrak_0\otimes_\mathbb. The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan. Real forms for Lie groups and algebraic groups 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. Classification Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by Satake diagrams, which are obtained from the Dynk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Compact Lie Algebra
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. Definition Formally, one may define a compact Lie algebra either as the Lie algebra of a compact Lie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree: * The Killing form on the Lie algebra of a compact Lie group is negative ''semi''definite, not negative definite in general. * If the Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compact ''semisimple'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups, including orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Compact Lie Algebra
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. Definition Formally, one may define a compact Lie algebra either as the Lie algebra of a compact Lie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree: * The Killing form on the Lie algebra of a compact Lie group is negative ''semi''definite, not negative definite in general. * If the Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compact ''semisimple'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Split Real Form
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0: : \mathfrak\simeq\mathfrak_0\otimes_\mathbb. The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan. Real forms for Lie groups and algebraic groups 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. Classification Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by Satake diagrams, which are obtained from the Dynk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Lie Theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map (Lie theory), exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter group, one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and Root datu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Semisimple Lie Algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, ideals.) Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of Characteristic (algebra), characteristic 0. For such a Lie algebra \mathfrak g, if nonzero, the following conditions are equivalent: *\mathfrak g is semisimple; *the Killing form \kappa(x, y) = \operatorname(\operatorname(x)\operatorname(y)) is non-degenerate; *\mathfrak g has no non-zero abelian ideals; *\mathfrak g has no non-zero solvable Lie algebra, solvable ideals; * the Radical of a Lie algebra, radical (maximal solvable ideal) of \mathfrak g is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Complex Numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |