1.
Exponential map (Riemannian geometry)
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In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold M to M itself. The Riemannian metric determines a canonical affine connection, and the map of the Riemannian manifold is given by the exponential map of this connection. Let M be a manifold and p a point of M. An affine connection on M allows one to define the notion of a geodesic through the point p, let v ∈ TpM be a tangent vector to the manifold at p. Then there is a unique geodesic γv satisfying γv = p with initial tangent vector γ′v = v, the corresponding exponential map is defined by expp = γv. In general, the map is only locally defined, that is, it only takes a small neighborhood of the origin at TpM. This is because it relies on the theorem of existence and uniqueness for ordinary differential equations which is local in nature, an affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle. Intuitively speaking, the map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and goes in that direction. Since v corresponds to the velocity vector of the geodesic, the distance traveled will be dependent on that. We can also reparametrize geodesics to be speed, so equivalently we can define expp = β where β is the unit-speed geodesic going in the direction of v. The Hopf–Rinow theorem asserts that it is possible to define the map on the whole tangent space if. In particular, compact manifolds are geodesically complete, however even if expp is defined on the whole tangent space, it will in general not be a global diffeomorphism. The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically via expp is called the injectivity radius of M at p. The cut locus of the map is, roughly speaking. This means, in particular, that the sphere of a small ball about the origin in TpM is orthogonal to the geodesics in M determined by those vectors. This motivates the definition of normal coordinates on a Riemannian manifold. Via the exponential map, it now can be defined as the Gaussian curvature of a surface through p determined by the image under expp of a 2-dimensional subspace of TpM. In general, Lie groups do not have a bi-invariant metric, take the example that gives the honest exponential map
2.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. 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, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
3.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an 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 and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, 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 μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. 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, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
4.
Classical group
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Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the classical groups. The finite analogues of the groups are the classical groups of Lie type. The term classical group was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups, the classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics, a few examples are the following. The rotation group SO is a symmetry of Euclidean space and all laws of physics. The special unitary group SU is the group of quantum chromodynamics. The classical groups are exactly the general linear groups over R, C and H together with the groups of non-degenerate forms discussed below. These groups are usually restricted to the subgroups whose elements have determinant 1. The classical groups, with the determinant 1 condition, are listed in the table below, in the sequel, the determinant 1 condition is not used consistently in the interest of greater generality. The complex classical groups are SL, SO and Sp, a group is complex according to whether its Lie algebra is complex. The real classical groups refers to all of the classical groups since any Lie algebra is a real algebra, the compact classical groups are the compact real forms of the complex classical groups. These are, in turn, SU, SO and Sp, one characterization of the compact real form is in terms of the Lie algebra g. If g = u + iu, the complexification of u, then if the connected group K generated by exp, X ∈ u is a compact, the classical groups can uniformly be characterized in a different way using real forms. The classical groups are the following, The complex linear algebraic groups SL, SO, for instance, SO∗ is a real form of SO, SU is a real form of Sl, and Sl is a real form of SO. Without the determinant 1 condition, replace the special linear groups with the general linear groups in the characterization. The algebraic groups in question are Lie groups, but the algebraic qualifier is needed to get the notion of real form. The classical groups are defined in terms of forms defined on Rn, Cn, and Hn, the quaternions, H, do not constitute a field because multiplication does not commute, they form a division ring or a skew field or non-commutative field
5.
General linear group
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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two matrices is again invertible, and the inverse of an invertible matrix is invertible. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix, for example, the general linear group over R is the group of n×n invertible matrices of real numbers, and is denoted by GLn or GL. More generally, the linear group of degree n over any field F, or a ring R, is the set of n×n invertible matrices with entries from F. Typical notation is GLn or GL, or simply GL if the field is understood, more generally still, the general linear group of a vector space GL is the abstract automorphism group, not necessarily written as matrices. The special linear group, written SL or SLn, is the subgroup of GL consisting of matrices with a determinant of 1, the group GL and its subgroups are often called linear groups or matrix groups. These groups are important in the theory of representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general. The modular group may be realised as a quotient of the linear group SL. If n ≥2, then the group GL is not abelian, if V has finite dimension n, then GL and GL are isomorphic. The isomorphism is not canonical, it depends on a choice of basis in V, in a similar way, for a commutative ring R the group GL may be interpreted as the group of automorphisms of a free R-module M of rank n. One can also define GL for any R-module, but in general this is not isomorphic to GL, over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore, a definition of GL is as the group of matrices with nonzero determinant. Over a non-commutative ring R, determinants are not at all well behaved, in this case, GL may be defined as the unit group of the matrix ring M. The general linear group GL over the field of numbers is a real Lie group of dimension n2. To see this, note that the set of all n×n real matrices, Mn, the subset GL consists of those matrices whose determinant is non-zero. The determinant is a map, and hence GL is an open affine subvariety of Mn. The Lie algebra of GL, denoted g l n, consists of all n×n real matrices with the serving as the Lie bracket. As a manifold, GL is not connected but rather has two connected components, the matrices with positive determinant and the ones with negative determinant, the identity component, denoted by GL+, consists of the real n×n matrices with positive determinant
6.
Special linear group
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In mathematics, the special linear group 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 subgroup of the general linear group given by the kernel of the determinant det. Where we write F× for the group of F. These elements are special in that fall on a subvariety of the general linear group – they satisfy a polynomial equation. When F is R or C, SL is a Lie subgroup of GL of dimension n2 −1, the Lie algebra s l of SL consists of all n × n matrices over F with vanishing trace. The Lie bracket is given by the commutator, any invertible matrix can be uniquely represented according to the polar decomposition as the product of a unitary matrix and a hermitian matrix with positive eigenvalues. Therefore, a linear matrix can be written as the product of a special unitary matrix. Thus the topology of the group SL is the product of the topology of SU, the topology of SL is the product of the topology of SO and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be expressed as the exponential of symmetric traceless matrices. The group SL, like SU, is simply connected while SL, SL has the same fundamental group as GL+ or SO, that is, Z for n =2 and Z2 for n >2. Two related subgroups, which in some cases coincide with SL, and in cases are accidentally conflated with SL, are the commutator subgroup of GL. These are both subgroups of SL, but in general do not coincide with it, the group generated by transvections is denoted E or TV. By the second Steinberg relation, for n ≥3, transvections are commutators, however, if A is a field with more than 2 elements, then E =, and if A is a field with more than 3 elements, E =. For more general rings the stable difference is measured by the special Whitehead group SK1, = SL/E, if working over a ring where SL is generated by transvections, one can give a presentation of SL using transvections with some relations. A sufficient set of relations for SL for n ≥3 is given by two of the Steinberg relations, plus a third relation, let Tij, = eij be the elementary matrix with 1s on the diagonal and in the ij position, and 0s elsewhere. Then = T i k for i ≠ k =1 for i ≠ l, j ≠ k 4 =1 are a set of relations for SL. The group GL splits over its determinant, and therefore GL can be written as a product of SL by F×, GL = SL ⋊ F×. 2307/2159559, JSTOR2159559
7.
Orthogonal group
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Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O is the orthogonal group, denoted SO. This group is called the rotation group, because, in dimensions 2 and 3. In low dimension, these groups have been studied, see SO, SO and SO. This is a subgroup of the linear group GL given by O = where QT is the transpose of Q and I is the identity matrix. This article mainly discusses the groups of quadratic forms that may be expressed over some bases as the dot product, over the reals. Over the reals, for any quadratic form, there is a basis. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O, for details, see indefinite orthogonal group. The derived subgroup Ω of O is an often studied object because, the Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form. The determinant of any orthogonal matrix is either 1 or −1, the orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O known as the special orthogonal group SO, consisting of all proper rotations. By analogy with GL–SL, the group is sometimes called the general orthogonal group and denoted GO. The term rotation group can be used to either the special or general orthogonal group. When this distinction is to be emphasized, the groups may be denoted O and O, reserving n for the dimension of the space. The letters p or r are also used, indicating the rank of the corresponding Lie algebra, in odd dimension the corresponding Lie algebra is s o, while in even dimension the Lie algebra is s o. In two dimensions, O is the group of all rotations about the origin and all reflections along a line through the origin, SO is the group of all rotations about the origin. These groups are related, SO is a subgroup of O of index 2. More generally, in any number of dimensions an even number of reflections gives a rotation, therefore, the rotations define a subgroup of O, but the reflections do not define a subgroup. A reflection through the origin may be generated as a combination of one reflection along each of the axes, the reflection through the origin is not a reflection in the usual sense in even dimensions, but rather a rotation
8.
Special orthogonal group
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Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O is the orthogonal group, denoted SO. This group is called the rotation group, because, in dimensions 2 and 3. In low dimension, these groups have been studied, see SO, SO and SO. This is a subgroup of the linear group GL given by O = where QT is the transpose of Q and I is the identity matrix. This article mainly discusses the groups of quadratic forms that may be expressed over some bases as the dot product, over the reals. Over the reals, for any quadratic form, there is a basis. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O, for details, see indefinite orthogonal group. The derived subgroup Ω of O is an often studied object because, the Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form. The determinant of any orthogonal matrix is either 1 or −1, the orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O known as the special orthogonal group SO, consisting of all proper rotations. By analogy with GL–SL, the group is sometimes called the general orthogonal group and denoted GO. The term rotation group can be used to either the special or general orthogonal group. When this distinction is to be emphasized, the groups may be denoted O and O, reserving n for the dimension of the space. The letters p or r are also used, indicating the rank of the corresponding Lie algebra, in odd dimension the corresponding Lie algebra is s o, while in even dimension the Lie algebra is s o. In two dimensions, O is the group of all rotations about the origin and all reflections along a line through the origin, SO is the group of all rotations about the origin. These groups are related, SO is a subgroup of O of index 2. More generally, in any number of dimensions an even number of reflections gives a rotation, therefore, the rotations define a subgroup of O, but the reflections do not define a subgroup. A reflection through the origin may be generated as a combination of one reflection along each of the axes, the reflection through the origin is not a reflection in the usual sense in even dimensions, but rather a rotation
9.
Unitary group
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In mathematics, the unitary group of degree n, denoted U, is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the linear group GL. Hyperorthogonal group is a name for the unitary group, especially over finite fields. In the simple case n =1, the group U corresponds to the circle group, all the unitary groups contain copies of this group. The unitary group U is a real Lie group of dimension n2, the Lie algebra of U consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. Since the determinant of a matrix is a complex number with norm 1. The kernel of this homomorphism is the set of matrices with determinant 1. This subgroup is called the special group, denoted SU . We then have an exact sequence of Lie groups,1 → SU → U → U →1. This short exact sequence splits so that U may be written as a product of SU by U . Here the U subgroup of U can be taken to consist of matrices, the unitary group U is nonabelian for n >1. The center of U is the set of scalar matrices λ I with λ ∈ U , the center is then isomorphic to U . Since the center of U is a 1 -dimensional abelian normal subgroup of U , the unitary group is not semisimple but is reductive. The unitary group U is endowed with the topology as a subset of M, the set of all n × n complex matrices. As a topological space, U is both compact and connected, the compactness of U follows from the Heine–Borel theorem and the fact that it is a closed and bounded subset of M. To show that U is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S, Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write A = S diag S −1, a path in U from the identity to A is then given by t ↦ S diag S −1. The unitary group is not simply connected, the group of U is infinite cyclic for all n, π1 ≅ Z
10.
Special unitary group
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In mathematics, the special unitary group of degree n, denoted SU, is the Lie group of n×n unitary matrices with determinant 1. The group operation is matrix multiplication, the special unitary group is a subgroup of the unitary group U, consisting of all n×n unitary matrices. As a compact group, U is the group that preserves the standard inner product on Cn. It is itself a subgroup of the linear group, SU ⊂ U ⊂ GL. The SU groups find application in the Standard Model of particle physics, especially SU in the electroweak interaction. The simplest case, SU, is the group, having only a single element. The group SU is isomorphic to the group of quaternions of norm 1, since unit quaternions can be used to represent rotations in 3-dimensional space, there is a surjective homomorphism from SU to the rotation group SO whose kernel is. SU is also identical to one of the groups of spinors, Spin. The special unitary group SU is a real Lie group and its dimension as a real manifold is n2 −1. Topologically, it is compact and simply connected, algebraically, it is a simple Lie group. The center of SU is isomorphic to the cyclic group Zn and its outer automorphism group, for n ≥3, is Z2, while the outer automorphism group of SU is the trivial group. A maximal torus, of rank n −1, is given by the set of matrices with determinant 1. The Weyl group is the symmetric group Sn, which is represented by signed permutation matrices, the Lie algebra of SU, denoted by su, can be identified with the set of traceless antihermitian n×n complex matrices, with the regular commutator as Lie bracket. Particle physicists often use a different, equivalent representation, the set of traceless hermitian n×n complex matrices with Lie bracket given by −i times the commutator, the Lie algebra su can be generated by n2 operators O ^ i j, i, j=1,2. N, which satisfy the commutator relationships = δ j k O ^ i ℓ − δ i ℓ O ^ k j for i, j, k, ℓ =1,2, N, where δjk denotes the Kronecker delta. Additionally, the operator N ^ = ∑ i =1 n O ^ i i satisfies =0, which implies that the number of independent generators of the Lie algebra is n2 −1. We also take ∑ c, e =1 n 2 −1 d a c e d b c e = n 2 −4 n δ a b as a normalization convention. In the -dimensional adjoint representation, the generators are represented by × matrices, whose elements are defined by the structure constants themselves, SU is the following group, S U =, where the overline denotes complex conjugation