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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Representation of a Lie group
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In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a tool in their study being the use of the corresponding infinitesimal representations of Lie algebras. The physics literature sometimes passes over the distinction between Lie groups and Lie algebras, let us first discuss representations acting on finite-dimensional complex vector spaces. A representation of a Lie group G on a complex vector space V is a smooth group homomorphism Ψ. For n-dimensional V, the group of V is identified with a subset of the complex square matrices of order n. The automorphism group of V is given the structure of a manifold using this identification. The condition that Ψ is smooth, in the definition above, if a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL. This is known as a matrix representation, a representation of a Lie group G on a vector space V is a smooth group homomorphism G→Aut from G to the automorphism group of V. If a basis for the vector space V is chosen, the representation can be expressed as a homomorphism into general linear group GL and this is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases for V and W. On the Lie algebra level, there is a linear mapping from the Lie algebra of G to End preserving the Lie bracket. See representation of Lie algebras for the Lie algebra theory, if the homomorphism is in fact a monomorphism, the representation is said to be faithful. A unitary representation is defined in the way, except that G maps to unitary matrices. If G is a compact Lie group, every representation is equivalent to a unitary one. This definition can handle representations on infinite-dimensional Hilbert spaces, such representations can be found in e. g. quantum mechanics, but also in Fourier analysis as shown in the following example. Let G=R, and let the complex Hilbert space V be L2 and we define the representation Ψ, R → B by Ψ → f. See also Wigners classification for representations of the Poincaré group, if G is a semisimple group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. The irreducibles are indexed by highest weight, the allowable highest weights satisfy a suitable positivity condition, the characters of the irreducible representations are given by the Weyl character formula

3.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars

4.
Lie algebra representation
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The notion is closely related to that of a representation of a Lie group. In the study of representations of a Lie algebra, a ring, called the universal enveloping algebra. The universality of this says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra. Explicitly, this means that ρ is a map that satisfies ρ = = ρ x ρ y − ρ y ρ x for all x, y in g. The vector space V, together with the representation ρ, is called a g -module, the representation ρ is said to be faithful if it is injective. One can equivalently define a g -module as a vector space V together with a map g × V → V such that ⋅ v = x ⋅ − y ⋅ for all x, y in g and v in V. This is related to the definition by setting x ⋅ v = ρx. The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra g on itself, ad, g → g l, x ↦ ad x, indeed, by virtue of the Jacobi identity, ad is a Lie algebra homomorphism. A Lie algebra representation also arises in nature. e, for example, let c g = g x g −1. Then the differential of c g, G → G at the identity is an element of G L. Denoting it by Ad one obtains a representation Ad of G on the vector space g and this is the adjoint representation of G. Applying the preceding, one gets the Lie algebra representation d Ad and it can be shown that d e Ad = ad, the adjoint representation of g. Then a linear map f, V → W is a homomorphism of g -modules if it is g -equivariant, if f is bijective, V, W are said to be equivalent. Similarly, many other constructions from module theory in abstract algebra carry over to this setting, submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc. Then V is said to be semisimple or completely reducible if it satisfies the following equivalent conditions, V is the sum of its simple submodules. Every submodule of V is a direct summand, for every submodule W of V, if g is a finite-dimensional semisimple Lie algebra over a field of characteristic zero and V is finite-dimensional, then V is semisimple. A Lie algebra is said to be if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive, an element v of V is said to be g -invariant if x v =0 for all x ∈ g. The set of all invariant elements is denoted by V g, V ↦ V g is a left-exact functor