Orthogonal group - Citizendia

In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix This is a subgroup of the general linear group GL(n,F) given by

$\mathrm{O}(n,F) = \{ Q \in \mathrm{GL}(n,F) \mid Q^T Q = Q Q^T = I \}.$

where Q^T is the transpose of Q. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a The classical orthogonal group over the real numbers is usually just written O(n).

More generally the orthogonal group of a non-singular quadratic form over F is the group of matrices preserving the form. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables The Cartan-Dieudonné theorem describes the structure of the orthogonal group. In Mathematics, the Cartan-Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, is a Theorem on the structure of the Automorphism

Every orthogonal matrix has determinant either 1 or −1. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n,F) known as the special orthogonal group SO(n,F). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. If the characteristic of F is 2, then 1 = −1, hence O(n,F) and SO(n,F) coincide; otherwise the index of SO(n,F) in O(n,F) is 2. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH In characteristic 2 and even dimension, many authors define the SO(n,F) differently as the kernel of the Dickson invariant; then it usually has index 2 in O(n,F).

Both O(n,F) and SO(n,F) are algebraic groups, because the condition that a matrix be orthogonal, i. In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by-

Groups

Group theory
This box: view • talk • edit

Over the real number field

Over the field R of real numbers, the orthogonal group O(n,R) and the special orthogonal group SO(n,R) are often simply denoted by O(n) and SO(n) if no confusion is possible. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, the real numbers may be described informally in several different ways They form real compact Lie groups of dimension n(n-1)/2. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it O(n,R) has two connected components, with SO(n,R) being the identity component, i. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, the identity component of a Topological group G is the connected component G 0 that contains the Identity e. , the connected component containing the identity matrix. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main

The real orthogonal and real special orthogonal groups have the following geometric interpretations

O(n,R) is a subgroup of the Euclidean group E(n), the group of isometries of Rⁿ; it contains those which leave the origin fixed. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional For the Mechanical engineering and Architecture usage see Isometric projection. It is the symmetry group of the sphere (n = 3) or hypersphere and all objects with spherical symmetry, if the origin is chosen at the center. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension.

SO(n,R) is a subgroup of E⁺(n), which consists of direct isometries, i. e. , isometries preserving orientation; it contains those which leave the origin fixed. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

{ I, −I } is a normal subgroup and even a characteristic subgroup of O(n,R), and, if n is even, also of SO(n,R). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, a characteristic subgroup of a group G is a Subgroup H that is invariant under each Automorphism of If n is odd, O(n,R) is the direct product of SO(n,R) and { I, −I }. In Mathematics, one can often define a direct product of objectsalready known giving a new one The cyclic group of k-fold rotations C_k is for every positive integer k a normal subgroup of O(2,R) and SO(2,R). In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

Relative to suitable orthogonal bases, the isometries are of the form:

$\begin{bmatrix} \begin{matrix}R_1 & & \\ & \ddots & \\ & & R_k\end{matrix} & 0 \\ 0 & \begin{matrix}\pm 1 & & \\ & \ddots & \\ & & \pm 1\end{matrix} \\ \end{bmatrix}$

where the matrices R₁,. . . ,R_k are 2-by-2 rotation matrices.

The symmetry group of a circle is O(2,R), also called Dih(S¹), where S¹ denotes the multiplicative group of complex numbers of absolute value 1. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

SO(2,R) is isomorphic (as a Lie group) to the circle S¹ (circle group). In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) to the orthogonal matrix

$\begin{bmatrix}\cos(\phi)&-\sin(\phi)\\ \sin(\phi)&\cos(\phi)\end{bmatrix}.$

The group SO(3,R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering. See rotation group and the general formula for a 3 × 3 rotation matrix in terms of the axis and the angle. This article is about rotations in three-dimensional Euclidean space In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed

In terms of algebraic topology, for n > 2 the fundamental group of SO(n,R) is cyclic of order 2, and the spinor group Spin(n) is its universal cover. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics the spin group Spin( n) is the double cover of the Special orthogonal group SO( n) such that there exists a Short In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the spinor group Spin(2) is the unique 2-fold cover). In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

The Lie algebra associated to the Lie groups O(n,R) and SO(n,R) consists of the skew-symmetric real n-by-n matrices, with the Lie bracket given by the commutator. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Lie bracket can refer to Lie algebra Lie bracket of vector fields In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. This Lie algebra is often denoted by o(n,R) or by so(n,R).

3D isometries which leave the origin fixed

The isometries of R³ which leave the origin fixed, forming the group O(3,R), can be categorized as follows:

SO(3,R):
- identity
- rotation about an axis through the origin by an angle not equal to 180°
- rotation about an axis through the origin by an angle of 180°
the same with inversion in the origin (x is mapped to −x), i. In Euclidean geometry, the inversion of a point X in respect to a point P is a point X * such that P is the midpoint of e. respectively:
- inversion in the origin
- rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin which is perpendicular to the axis
- reflection in a plane through the origin

The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations. In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or

See also the similar overview including translations. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional

Conformal group

Main article: Conformal group

Being isometries (preserving distances), orthogonal transforms also preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane The group of conformal linear maps of Rⁿ is denoted CO(n), and consists of the product of the orthogonal group with the group of dilations. In Mathematics, a dilation is a function &fnof from a Metric space into itself that satisfies the identity d(f(xf(y=rd(xy \ If n is odd, these two subgroups do not intersect, and they are a direct product: $\operatorname{CO}(2n+1) = \operatorname{O}(2n+1) \times \mathbf{R}$ , while if n is even, these subgroups intersect in $\pm 1$ , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: $\operatorname{CO}(2n) = \operatorname{O}(2n) \times \mathbf{R}^+$ . In Mathematics, one can often define a direct product of objectsalready known giving a new one

Similarly one can define CSO(n); note that this is always : $\operatorname{CSO}(n) := \operatorname{CO}(n) \cap \operatorname{GL}_+(n) = \operatorname{SO}(n) \times \mathbf{R}^+$ .

Over the complex number field

Over the field C of complex numbers, O(n,C) and SO(n,C) are complex Lie groups of dimension n(n-1)/2 over C (which means the dimension over R is twice that). Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted O(n,C) has two connected components, and SO(n,C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact.

Just as in the real case SO(n,C) is not simply connected. For n > 2 the fundamental group of SO(n,C) is cyclic of order 2 whereas the fundamental group of SO(2,C) is infinite cyclic. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

The complex Lie algebra associated to O(n,C) and SO(n,C) consists of the skew-symmetric complex n-by-n matrices, with the Lie bracket given by the commutator. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Lie bracket can refer to Lie algebra Lie bracket of vector fields In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative.

Topology

Low dimensional

The low dimensional (real) orthogonal groups are familiar spaces:

$\begin{align} O(0) &= \left\{1\right\} = *\\ O(1) &= \left\{\pm 1\right\} = S^0\\ SO(2) &= S^1\\ SO(3) &= \mathbf{RP}^3 \end{align}$

Homotopy groups

The homotopy groups of the orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. In the mathematical field of Algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other

However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions

$O(0) \subset O(1)\subset O(2)\subset\cdots\subset O = \bigcup_{k=0}^\infty O(k)$

(as the inclusions are all closed inclusions, hence cofibrations, this can also be interpreted as a union). In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In Mathematics, in particular Homotopy theory, a Continuous mapping i\colon A \to X where A and X

$S n$ is a homogeneous space for $O (n + 1)$ , and one has the following fiber bundle:

$O(n) \to O(n+1) \to S^n,$

which can be understood as "The orthogonal group $O (n + 1)$ acts transitively on the unit sphere $S n$ , and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower". In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. The map $O(n) \to O(n+1)$ is the natural inclusion.

Thus the inclusion $O(n) \to O(n+1)$ is (n-1)-connected, so the homotopy groups stabilize, and $π k (O) = π k (O (n))$ for $n > k + 1$ : thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces. n-connected redirects here for the concept in graph theory see Connectivity (graph theory.

Via Bott periodicity, $\Omega^8 O \simeq O$ , thus the homotopy groups of O are 8-fold periodic, meaning $π k + 8 O = π k O$ , and one need only compute the lower 8 homotopy groups to compute them all. In Mathematics, the Bott periodicity theorem is a result from Homotopy theory discovered by Raoul Bott during the latter part of the 1950s which proved

$\begin{align} \pi_0 O &= \mathbf Z/2\\ \pi_1 O &= \mathbf Z/2\\ \pi_2 O &= 0\\ \pi_3 O &= \mathbf Z\\ \pi_4 O &= 0\\ \pi_5 O &= 0\\ \pi_6 O &= 0\\ \pi_7 O &= \mathbf Z\\ \end{align}$

Relation to KO-theory

Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: $π k O = π k + 1 B O$ . In Mathematics, topological K-theory is a branch of Algebraic topology. In Topology, a branch of mathematics the clutching construction is a way of constructing fiber bundles particularly vector bundles on spheres

Setting $KO = BO \times \mathbf Z = \Omega^{-1} O \times \mathbf Z$ (to make $π 0$ fit into the periodicity), one obtains:

$\begin{align} \pi_0 KO &= \mathbf Z\\ \pi_1 KO &= \mathbf Z/2\\ \pi_2 KO &= \mathbf Z/2\\ \pi_3 KO &= 0\\ \pi_4 KO &= \mathbf Z\\ \pi_5 KO &= 0\\ \pi_6 KO &= 0\\ \pi_7 KO &= 0\\ \end{align}$

Computation and Interpretation of homotopy groups

Low-dimensional groups

The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.

$\pi_0(O) = \pi_0(O(1)) = \mathbf Z/2$ from orientation-preserving/reversing (this class survives to $O (2)$ and hence stably)

$SO(3) = \mathbf{RP}^3 = S^3/(\mathbf Z/2)$ yields

$\pi_1(O) = \pi_1(SO(3)) = \mathbf Z/2$ which is spin
$π 2 (O) = π 2 (S O (3)) = 0$ , which surjects onto $π 2 (S O (4))$ ; this latter thus vanishes

Lie groups

From general facts about Lie groups, $π 2 G$ always vanishes, and $π 3 G$ is free (free abelian). See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which In Mathematics the spin group Spin( n) is the double cover of the Special orthogonal group SO( n) such that there exists a Short In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Vector bundles

From the vector bundle point of view, $π 0 (K O)$ is vector bundles over $S 0$ , which is two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so

$\pi_0(KO) = \mathbf Z$ is dimension

Loop spaces

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret higher homotopy of O as lower homotopy of simple to analyze spaces. In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, the Bott periodicity theorem is a result from Homotopy theory discovered by Raoul Bott during the latter part of the 1950s which proved Using $π 0$ , O and O/U have two components, $KO = BO \times \mathbf Z$ and $KSp = BSp \times \mathbf Z$ have $\mathbf Z$ components, and the rest are connected.

Interpretation of homotopy groups

In a nutshell:^[1]

$\pi_0(KO) = \mathbf Z$ is dimension
$\pi_1(KO) = \mathbf Z/2$ is orientation
$\pi_2(KO) = \mathbf Z/2$ is spin
$\pi_4(KO) = \mathbf Z$ is topological quantum field theory

Let $F = \mathbf R, \mathbf C, \mathbf H, \mathbf O$ , and let $L F$ be the tautological line bundle over the projective line $\mathbf{FP}^1$ , and $[L F]$ its class in K-theory. In Mathematics, the dimension of a Vector space V is the cardinality (i See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which In Mathematics the spin group Spin( n) is the double cover of the Special orthogonal group SO( n) such that there exists a Short A topological quantum field theory (or topological field theory or TQFT) is a Quantum field theory which computes Topological invariants Noting that $\mathbf{RP}^1 = S^1, \mathbf{CP}^1 = S^2, \mathbf{HP}^1 = S^4, \mathbf{OP}^1 = S^8$ , these yield vector bundles over the corresponding spheres, and

$π 1 (K O)$ is generated by $[L_{\mathbf R}]$
$π 2 (K O)$ is generated by $[L_{\mathbf C}]$
$π 4 (K O)$ is generated by $[L_{\mathbf H}]$
$π 8 (K O)$ is generated by $[L_{\mathbf O}]$

Over finite fields

Orthogonal groups can also be defined over finite fields $\mathbf{F}_q$ , where $q$ is a power of a prime $p$ . When defined over such fields, they come in two types in even dimension: $O + (2 n, q)$ and $O - (2 n, q)$ ; and one type in odd dimension: $O (2 n + 1, q)$ .

If $V$ is the vector space on which the orthogonal group $G$ acts, it can be written as a direct orthogonal sum as follows:

$V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W$ ,

where $L i$ are hyperbolic lines and $W$ contains no singular vectors. If $W = 0$ , then $G$ is of plus type. If $W = < w >$ then $G$ has odd dimension. If $W$ has dimension 2, $G$ is of minus type.

In the special case where n = 1, $O ε (2, q)$ is a dihedral group of order $2(q - ε)$ . In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections

We have the following formulas for the order of these groups, O(n,q) = { A in GL(n,q) : A·A^t=I }, when the characteristic is greater than two.

$|O(2n+1,q)|=2q^n\prod_{i=0}^{n-1}(q^{2n}-q^{2i})$

If $- 1$ is a square in $\mathbf{F}_q$

$|O(2n,q)|=2(q^n-1)\prod_{i=1}^{n-1}(q^{2n}-q^{2i})$

If $- 1$ is a nonsquare in $\mathbf{F}_q$

$|O(2n,q)|=2(q^n+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i})$

The Dickson invariant

For orthogonal groups in even dimensions, the Dickson invariant is a homomorphism from the orthogonal group to Z/2Z, and is 0 or 1 depending on whether an element is the product of an even or odd number of reflections. Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives extra information. In characteristic 2 many authors define the special orthogonal group to be the elements of Dickson invariant 0, rather than the elements of determinant 1.

The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way (in all dimensions). In Mathematics, Clifford algebras are a type of Associative algebra.

Orthogonal groups of characteristic 2

Over fields of characteristic 2 orthogonal groups often behave differently. This section lists some of the differences.

Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4 dimensional over the field with 2 elements and the Witt index is 2 (Grove 2002, Theorem 6. 6 and 14. 16). Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthgonal to a vector u takes a vector v to v+B(v,u)/Q(u)·u where B is the bilinear form and Q is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes v to v-2·B(v,u)/Q(u)·u. In Mathematics, a Householder transformation in 3-dimensional space is the reflection of a vector in a plane.

The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2.

In odd dimensions 2n+1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group.

In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.

The spinor norm

The spinor norm is a homomorphism from an orthogonal group over a field F to

F^*/F^*2,

the multiplicative group of the field F up to square elements, that takes reflection in a vector of norm n to the image of n in F^*/F^*2. In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose

For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.

Galois cohomology and orthogonal groups

In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. In Mathematics, Galois cohomology is the study of the Group cohomology of Galois modules that is the application of Homological algebra to In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomena is concerned. The first point is that quadratic forms over a field can be identified as a Galois H¹, or twisted forms (torsors) of an orthogonal group. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, a principal homogeneous space, or torsor, for a group G is a set X on which G acts freely and As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant. In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the

The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). In Mathematics the spin group Spin( n) is the double cover of the Special orthogonal group SO( n) such that there exists a Short This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). In Mathematics, Clifford algebras are a type of Associative algebra. The spin covering of the orthogonal group provides a short exact sequence of algebraic groups. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse

$1 \rightarrow \mu_2 \rightarrow Pin_V \rightarrow O_V \rightarrow 1$

Here μ₂ is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak The connecting homomorphism from H⁰(O_V) which is simply the group O_V(F) of F-valued points, to H¹(μ₂) is essentially the spinor norm, because H¹(μ₂) is isomorphic to the multiplicative group of the field modulo squares. In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the

There is also the connecting homomorphism from H¹ of the orthogonal group, to the H² of the kernel of the spin covering. The cohomology is non-abelian, so that this is as far as we can go, at least with the conventional definitions.

Important subgroups

In physics, particular in the areas of Kaluza-Klein compacification, it is important to find out the subgroups of the orthogonal group. The main ones are:

$O(n) \supset O(n-1)$

$O(2n) \supset SU(n)$

$O(2n) \supset USp(n)$

$O(7) \supset G_2$

The orthogonal group O(n) is also an important subgroup of various lie groups:

$SU(n) \supset O(n)$

$USp(2n) \supset O(n)$

$G_2 \supset O(3)$

$F_4 \supset O(9)$

$E_6 \supset O(10)$

$E_7 \supset O(12)$

$E_8 \supset O(16)$

The group O(10) is of special importance in superstring theory because it is the symmetry group of 10 dimensional space-time. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group See also String theory Superstring theory is an attempt to explain all of the particles and Fundamental forces of nature in one theory by modelling

Footnotes

^ John Baez "This Week's Finds in Mathematical Physics" week 105

References

Grove, Larry C. (2002), Classical groups and geometric algebra, vol. 39, Graduate Studies in Mathematics, Providence, R. I. : American Mathematical Society, MR 1859189, ISBN 978-0-8218-2019-3

External links

The American Mathematical Society (AMS is an association of professional Mathematicians dedicated to the interests of mathematical research and scholarship which Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many This article is about rotations in three-dimensional Euclidean space In Mathematics, SO(8 is the Special orthogonal group acting on eight-dimensional Euclidean space. In Mathematics, the indefinite orthogonal group, O( p, q) is the Lie group of all Linear transformations of a n = p In Mathematics, the unitary group of degree n, denoted U( n) is the group of n × n unitary matrices In Mathematics, the name symplectic group can refer to two different but closely related types of mathematical groups. In Mathematics, the Classification of finite simple groups states thatevery finite Simple group is cyclic, or alternating, or in one of 16 families In Mathematics, the Simple Lie groups were classified by Élie Cartan.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents