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Movie Title Year Distributor Notes Rev Formats All in One 2013 realitykings.com Anal Facial Bald DO Brazil Xposed 2013 Evil Angel Anal Facial 1 DRO Bunda Mais Linda do Ano ... A Deliciosa Keyte Bittencourt 2013 turmadosexo.com.br Carioca 2013 sexo.uol.com.br Carnaval 2014 2014 HardSexy Elias 2013 sexo.uol.com.br Felipe 2013 sexo.uol.com.br Igor 2013 sexo.uol.com.br Jonis 2013 sexo.uol.com.br Juan 2013 sexo.uol.com.br Latina Anal Hunt 2 2016 Hot Hot Films O Marcelo 2013 sexo.uol.com.br Nem Pensou Antes de Cair de Boca 2013 redesexo.com.br Facial Plugged 3 2017 West Sex Brazil O Rafael 2013 sexo.uol.com.br In mathematics, a Lie group (pronounced /li?/ "Lee") is a group whose elements are organized continuously and smoothly, as opposed to discrete groups, where the elements are separated—this makes Lie groups differentiable manifolds. Classically, such groups were found by studying matrix subgroups {\displaystyle G}G contained in {\displaystyle {\text{GL}}_{n}(\mathbb {R} )}{\displaystyle {\text{GL}}_{n}(\mathbb {R} )} or {\displaystyle {\text{GL}}_{n}(\mathbb {C} )}{\displaystyle {\text{GL}}_{n}(\mathbb {C} )}, the group of {\displaystyle n\times n}n\times n invertible matrices over {\displaystyle \mathbb {R} }\mathbb {R} or {\displaystyle \mathbb {C} }\mathbb {C} . Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups.
In rough terms, a Lie group is a continuous group: it is a group whose elements are described by several real parameters. As such, Lie groups provide a natural model for the concept of continuous symmetry, such as rotational symmetry in three dimensions (given by the special orthogonal group {\displaystyle {\text{SO}}(3)}{\displaystyle {\text{SO}}(3)}). Lie groups are widely used in many parts of modern mathematics and physics. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations. Contents 1 Overview 2 Definitions and examples 2.1 First examples 2.2 Non-example 2.3 Matrix Lie groups 2.4 Related concepts 2.5 Topological definition 3 More examples of Lie groups 3.1 Dimensions one and two 3.2 Additional examples 3.3 Constructions 3.4 Related notions 4 Basic concepts 4.1 The Lie algebra associated with a Lie group 4.2 Homomorphisms and isomorphisms 4.3 Lie group versus Lie algebra isomorphisms 4.4 Simply connected Lie groups 4.5 The exponential map 4.6 Lie subgroup 5 Representations 6 Early history 7 The concept of a Lie group, and possibilities of classification 8 Infinite-dimensional Lie groups 9 See also 10 Notes 10.1 Explanatory notes 10.2 Citations 11 References

Overview The set of all complex numbers with absolute value 1 (corresponding to points on the circle of center 0 and radius 1 in the complex plane) is a Lie group under complex multiplication: the circle group. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra. Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold. Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the representations of the Lie group (or of its Lie algebra) are especially important. Representation theory is used extensively in particle physics. Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory. In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; p-adic Lie groups play an important role, via their connections with Galois representations in number theory. Definitions and examples A real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication {\displaystyle \mu :G\times G\to G\quad \mu (x,y)=xy}\mu :G\times G\to G\quad \mu (x,y)=xy means that µ is a smooth mapping of the product manifold G × G into G. These two requirements can be combined to the single requirement that the mapping {\displaystyle (x,y)\mapsto x^{-1}y}(x,y)\mapsto x^{-1}y be a smooth mapping of the product manifold into G. First examples The 2×2 real invertible matrices form a group under multiplication, denoted by GL(2, R) or by GL2(R): {\displaystyle \operatorname {GL} (2,\mathbf {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\det A=ad-bc\neq 0\right\}.}\operatorname {GL} (2,\mathbf {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\det A=ad-bc\neq 0\right\}. This is a four-dimensional noncompact real Lie group; it is an open subset of {\displaystyle \mathbb {R} ^{4}}{\mathbb R}^{4}. This group is disconnected; it has two connected components corresponding to the positive and negative values of the determinant. The rotation matrices form a subgroup of GL(2, R), denoted by SO(2, R). It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle. Using the rotation angle {\displaystyle \varphi }\varphi as a parameter, this group can be parametrized as follows: {\displaystyle \operatorname {SO} (2,\mathbf {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\varphi \in \mathbf {R} /2\pi \mathbf {Z} \right\}.}\operatorname {SO} (2,\mathbf {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\varphi \in \mathbf {R} /2\pi \mathbf {Z} \right\}. Addition of the angles corresponds to multiplication of the elements of SO(2, R), and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps. The affine group of one dimension is a two-dimensional matrix Lie group, consisting of {\displaystyle 2\times 2}2\times 2 real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form {\displaystyle A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} .}{\displaystyle A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} .} Non-example We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by {\displaystyle H=\left\{\left.\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right)\right|\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left.\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right)\right|\theta ,\phi \in \mathbb {R} \right\},}{\displaystyle H=\left\{\left.\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right)\right|\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left.\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right)\right|\theta ,\phi \in \mathbb {R} \right\},} with {\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} }{\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} } a fixed irrational number, is a subgroup of the torus {\displaystyle \mathbb {T} ^{2}}{\mathbb T}^{2} that is not a Lie group when given the subspace topology.[1] If we take any small neighborhood {\displaystyle U}U of a point {\displaystyle h}h in {\displaystyle H}H, for example, the portion of {\displaystyle H}H in {\displaystyle U}U is disconnected. The group {\displaystyle H}H winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of {\displaystyle \mathbb {T} ^{2}}{\mathbb T}^{2}. A portion of the group {\displaystyle H}H inside {\displaystyle \mathbb {T} ^{2}}{\mathbb T}^{2}. Small neighborhoods of the element {\displaystyle h\in H}h\in H are disconnected in the subset topology on {\displaystyle H}H The group {\displaystyle H}H can, however, be given a different topology, in which the distance between two points {\displaystyle h_{1},h_{2}\in H}{\displaystyle h_{1},h_{2}\in H} is defined as the length of the shortest path in the group {\displaystyle H}H joining {\displaystyle h_{1}}h_{1} to {\displaystyle h_{2}}h_{2}. In this topology, {\displaystyle H}H is identified homeomorphically with the real line by identifying each element with the number {\displaystyle \theta }\theta in the definition of {\displaystyle H}H. With this topology, {\displaystyle H}H is just the group of real numbers under addition and is therefore a Lie group. The group {\displaystyle H}H is an example of a "Lie subgroup" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts

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