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Lie Algebra
In mathematics, a Lie algebra
Lie algebra
(pronounced /liː/ "Lee") is a vector space g displaystyle mathfrak g together with a non-associative, alternating bilinear map g × g → g ; ( x , y ) ↦ [ x , y ] displaystyle mathfrak g times mathfrak g rightarrow mathfrak g ;(x,y)mapsto [x,y] , called the Lie bracket, satisfying the Jacobi identity. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group
Lie group
gives rise to a Lie algebra
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Index Of A Lie Algebra
Index
Index
may refer to:Contents1 Arts, entertainment, and media1.1 Fictional entities 1.2 Periodicals and news portals 1.3 Other arts, entertainment and media2 Business enterprises and events 3 Finance 4 Places 5 Publishing and library studies 6 Science, technology, and mathematics6.1 Computer science 6.2 Economics 6.3 Mathematics and statistics6.3.1 Algebra 6.3.2 Analysis 6.3.3 Statistics6.4 Other uses in science and technology7 Other uses 8 See alsoArts, entertainment, and media[edit] Fictional entities[edit]Index, a character from the Japanese light novel, anime and manga A Certain Mag
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Homogeneous Space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group
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Group Theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science
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Conformal Group
In mathematics, the conformal group of a space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important:The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V such that for all x in V there exists a scalar λ such that Q ( T x ) = λ 2 Q ( x ) displaystyle Q(Tx)=lambda ^ 2 Q(x) For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations.The conformal group of the sphere is generated by the inversions in circles
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Compact Lie Algebra
Algebra
Algebra
(from Arabic
Arabic
"al-jabr" literally meaning "reunion of broken parts"[1]) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols;[2] it is a unifying thread of almost all of mathematics.[3] As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra
Elementary algebra
is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics
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Split Lie Algebra
Algebra
Algebra
(from Arabic
Arabic
"al-jabr" literally meaning "reunion of broken parts"[1]) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols;[2] it is a unifying thread of almost all of mathematics.[3] As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra
Elementary algebra
is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics
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Simple Lie Algebra
Algebra
Algebra
(from Arabic
Arabic
"al-jabr" literally meaning "reunion of broken parts"[1]) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols;[2] it is a unifying thread of almost all of mathematics.[3] As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra
Elementary algebra
is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics
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Lie Bracket Of Vector Fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted [X, Y]. Conceptually, the Lie bracket [X, Y] is the derivative of Y along the flow generated by X. A generalization of the Lie bracket is the Lie derivative, which allows differentiation of any tensor field along the flow generated by X
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Circle Group
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers[1] T = z ∈ C : z = 1 . displaystyle mathbb T = zin mathbb C :z=1 . The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers. Since C× is abelian, it follows that T is as well. The circle group is also the group U(1) of 1×1 complex-valued unitary matrices; these act on the complex plane by rotation about the origin
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Euclidean Group
In mathematics, the Euclidean group
Euclidean group
E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements are the isometries associated with the Euclidean distance, and are called Euclidean isometries, Euclidean transformations or Rigid transformations. Euclidean isometries are classified into direct isometries and indirect isometries, an indirect isometry being an isometry that transforms any object into its mirror image
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Loop Group
In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise.Contents1 Definition 2 Examples 3 Notes 4 References 5 See alsoDefinition[edit] In its most general form a loop group is a group of mappings from a manifold M to a topological group G. More specifically,[1] let M = S1, the circle in the complex plane, and let LG denote the space of continuous maps S1 → G, i.e. L G = γ : S 1 → G γ ∈ C ( S 1 , G ) , displaystyle LG= gamma :S^ 1 to Ggamma in C(S^ 1 ,G) , equipped with the compact-open topology. An element of LG is called a loop in G. Pointwise multiplication of such loops gives LG the structure of a topological group
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Special Linear Group
In mathematics, the special linear group SL(n, F) of degree n over a field F is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant det : GL ⁡ ( n , F ) → F × . displaystyle det colon operatorname GL (n,F)to F^ times
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Unitary Group
In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U(n) is a real Lie group of dimension n2
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Restricted Root System
In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its dual can be identified. For symmetric spaces of noncompact type arising as homogeneous spaces of a semisimple Lie group, the restricted root system and its Weyl group are related to the Iwasawa decomposition of the Lie group. See also[edit]Satake diagramReferences[edit]Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics, 225, Springer, ISBN 0387211543  Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0821828487  Onishchik, A. L.; Vinberg, E. B. (1994), Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Encyclopaedia of Mathematical Sciences, 41, Springer, ISBN 9783540546832  Wolf, Joseph A
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Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.The image of a rectangular grid on a square under a diffeomorphism from the square onto itself.Contents1 Definition 2 Diffeomorphisms of subsets of manifolds 3 Local description 4 Examples4.1 Surface deformations5 Diffeomorphism
Diffeomorphism
group5.1 Topology 5.2 Lie algebra 5.3 Examples 5.4 Transitivity 5.5 Extensions of diffeomorphisms 5.6 Connectedness 5.7 Homotopy types6 Homeomorphism
Homeomorphism
and diffeomorphism 7 See also 8 Notes 9 ReferencesDefinition[edit] Given two manifolds M and N, a differentiable map f : M → N is called a diffeomorphism if it is a bijection and its inverse f−1 : N → M is differentiable as well
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