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Root System
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Plant
Plants are the eukaryotes that form the Kingdom (biology), kingdom Plantae; they are predominantly Photosynthesis, photosynthetic. This means that they obtain their energy from sunlight, using chloroplasts derived from endosymbiosis with cyanobacteria to produce sugars from carbon dioxide and water, using the green pigment chlorophyll. Exceptions are parasitic plants that have lost the genes for chlorophyll and photosynthesis, and obtain their energy from other plants or fungi. Most plants are multicellular organism, multicellular, except for some green algae. Historically, as in Aristotle's biology, the plant kingdom encompassed all living things that were not animals, and included algae and fungi. Definitions have narrowed since then; current definitions exclude fungi and some of the algae. By the definition used in this article, plants form the clade Viridiplantae (green plants), which consists of the green algae and the embryophytes or land plants (hornworts, liverworts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line. Most commonly, the ambient space is -dimensional Euclidean space, in which case the hyperplanes are the -dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion ( geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces. In non-Euclidean geometry, the ambient space might be the -dimensional sphere or hyperbolic space, or more generally a pseudo-Riemannian space form, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faithful Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner product on V. The Weyl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion. Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A2 Weyl Group (revised)
A, or a, is the first letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is '' a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, '' a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest known ancestor of A is ''aleph''—the first letter of the Phoenician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (discrete Subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of R''n'', this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood. The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of Grigory Margulis states that in most cases all lattices are obtained as arithmetic groups. Lattices are also well-studied in some other classes of groups, in particular groups associated to Kac–Moody algebras and automorphisms groups of regular trees (the latter are known as ''tree lattices''). Lattices a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System C2 (fixed)
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the surface of the soil, but roots can also be aerial or aerating, that is, growing up above the ground or especially above water. Function The major functions of roots are absorption of water, plant nutrition and anchoring of the plant body to the ground. Types of Roots (major rooting system) Plants exhibit two main root system types: ''taproot'' and ''fibrous'', with variations like adventitious, aerial, and buttress roots, each serving specific functions. Taproot System Characterized by a single, main root growing vertically downward, with smaller lateral roots branching off. Examples. Dandelions, carrots, and many dicot plants. Fibrous RootSystem Consists of a network of thin, branching roots that spread out from the base of the stem, l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System B2
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the surface of the soil, but roots can also be aerial root, aerial or aerating, that is, growing up above the ground or especially above water. Function The major functions of roots are absorption of water, plant nutrition and anchoring of the plant body to the ground. Types of Roots (major rooting system) Plants exhibit two main root system types: ''taproot'' and ''fibrous'', with variations like adventitious, aerial, and buttress roots, each serving specific functions. Taproot System Characterized by a single, main root growing vertically downward, with smaller lateral roots branching off. Examples. Dandelions, carrots, and many dicot plants. Fibrous RootSystem Consists of a network of thin, branching roots that spread out from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root System G2
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the surface of the soil, but roots can also be aerial or aerating, that is, growing up above the ground or especially above water. Function The major functions of roots are absorption of water, plant nutrition and anchoring of the plant body to the ground. Types of Roots (major rooting system) Plants exhibit two main root system types: ''taproot'' and ''fibrous'', with variations like adventitious, aerial, and buttress roots, each serving specific functions. Taproot System Characterized by a single, main root growing vertically downward, with smaller lateral roots branching off. Examples. Dandelions, carrots, and many dicot plants. Fibrous RootSystem Consists of a network of thin, branching roots that spread out from the base of the stem, l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
