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Orthogonal group

In mathematics, the orthogonal group O(n) is the Lie group consisting of all n \times n real matrices A such that A^T A = I_n, where I_n is the identity matrix and A^T denotes the transpose of A.^[1] These matrices represent linear transformations that preserve the Euclidean inner product, meaning they map orthonormal bases to orthonormal bases and maintain lengths and angles between vectors.^[1] The group operation is matrix multiplication, and the identity element is the identity matrix, with inverses given by transposes since A^{-1} = A^T.^[1] As a classical Lie group, O(n) is compact and has dimension n(n-1)/2, reflecting the degrees of freedom in choosing orthonormal bases up to signs.^[2] It is disconnected, with two connected components distinguished by the determinant: matrices with \det A = 1 form the special orthogonal group SO(n), which is connected and represents orientation-preserving transformations (rotations), while those with \det A = -1 include reflections.^[3] Over the complex numbers, the orthogonal group O(n, \mathbb{C}) similarly preserves a non-degenerate symmetric bilinear form and has two connected components, with the identity component SO(n, \mathbb{C}).^[4] Orthogonal groups play a central role in geometry and linear algebra, as their elements correspond to isometries of Euclidean space \mathbb{R}^n, preserving distances and thus the structure of the space.^[5] In three dimensions, SO(3) is particularly notable as the rotation group, essential for describing rigid body motions in physics and computer graphics.^[1] Applications extend to analytic geometry for coordinate transformations, numerical algorithms involving orthogonality constraints (such as QR decomposition), and Fourier analysis for expanding functions in orthonormal bases.^[6]^[7] More advanced uses appear in representation theory, where orthogonal groups classify symmetries in quantum mechanics and particle physics.^[8]

Definition

Notation and naming

The orthogonal group in n-dimensional Euclidean space, denoted O(n), comprises all linear transformations that preserve the standard dot product on \mathbb{R}^n.^[9] This notation emphasizes the group's role in maintaining orthogonality among vectors under the Euclidean metric.^[10] The subgroup consisting of those elements with determinant equal to 1 is known as the special orthogonal group, denoted SO(n).^[9] The naming convention arises from the transformations' preservation of angles and lengths, a direct consequence of conserving the dot product, which generalizes the geometric notion of perpendicularity.^[11] The term "orthogonal" originates from the Greek roots orthos (straight or right) and gonia (angle), literally meaning "right-angled," reflecting the preservation of right angles between vectors.^[12] Charles Hermite introduced the phrase "orthogonal matrix" in 1854 to describe such transformations in his work on quadratic forms.^[13] For spaces equipped with an indefinite quadratic form of signature (p, q), where p + q = n, the analogous group is denoted O(p, q), adapting the notation to the non-Euclidean metric.^[11] This variation highlights the flexibility of the framework beyond positive-definite cases.^[14]

Basic definition

In mathematics, the orthogonal group associated to a vector space V over a field k equipped with a non-degenerate quadratic form q: V \to k is defined as the group O(V, q) consisting of all invertible linear transformations T: V \to V such that q(Tv) = q(v) for every v \in V.^[15] This condition ensures that T preserves the quadratic form q, making O(V, q) the full group of isometries of the quadratic space (V, q).^[15] Equivalently, these transformations are the automorphisms of V that preserve the associated symmetric bilinear form B(u, v) = \frac{1}{2} [q(u + v) - q(u) - q(v)], assuming \mathrm{char}(k) \neq 2.^[16] Over the real numbers \mathbb{R}, when V = \mathbb{R}^n is equipped with the standard positive definite quadratic form q(x) = x \cdot x = \sum_{i=1}^n x_i^2, the orthogonal group O(n, \mathbb{R}) (often denoted simply O(n)) consists of all n \times n real matrices A satisfying A^\top A = I_n, where I_n is the n \times n identity matrix.^[9] Every such matrix A has determinant \det(A) = \pm 1, since \det(A^\top A) = \det(I_n) = 1 implies [\det(A)]^2 = 1.^[9] More generally, over any field k (with \mathrm{char}(k) \neq 2), if G is the Gram matrix representing the symmetric bilinear form B with respect to a basis of V, then the elements of the orthogonal group are the matrices A \in \mathrm{GL}_n(k) satisfying A^\top G A = G.^[16] This matrix equation captures the preservation of the quadratic form in coordinates.^[16]

Geometric interpretation

Euclidean geometry

The orthogonal group O(n) is the group of all linear isometries of the Euclidean space \mathbb{R}^n that fix the origin, consisting of transformations that preserve distances and angles between vectors.^[17] These isometries derive their preserving properties from maintaining the standard quadratic form x \mapsto x^T x, ensuring \|Ax\| = \|x\| and \langle Ax, Ay \rangle = \langle x, y \rangle for all x, y \in \mathbb{R}^n.^[16] As referenced in the basic definition, elements A \in O(n) satisfy A^T A = I, which directly implies norm preservation.^[18] The group O(n) splits into orientation-preserving and orientation-reversing isometries, with the former forming the connected component SO(n) of rotations and the latter comprising the "odd" elements with determinant -1, such as reflections.^[19] Rotations constitute the identity-connected component of O(n), while reflections and their compositions with rotations reverse orientation.^[20] In two dimensions, O(2) is generated by rotations about the origin and a single reflection, such as over the x-axis, yielding all orientation-preserving and -reversing linear isometries fixing the origin.^[21] For instance, the reflection matrix \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} combined with rotation matrices produces the full group. In three dimensions, O(3) includes rotations around any axis through the origin as well as improper rotations, which combine a proper rotation with a reflection (or equivalently, with central inversion).^[22] These improper elements, like a 180-degree rotation followed by reflection in a plane perpendicular to the axis, account for all orientation-reversing isometries in \mathbb{R}^3.^[23]

Special orthogonal group

The special orthogonal group, denoted SO(n), is defined as the subgroup of the orthogonal group O(n) consisting of all n \times n real orthogonal matrices A satisfying \det A = 1, that is,

\text{SO}(n) = \{ A \in \text{O}(n) \mid \det A = 1 \}.

^[24] This makes SO(n) the kernel of the surjective determinant homomorphism \det: \text{O}(n) \to \{\pm 1\}, establishing SO(n) as a normal subgroup of index 2 in O(n).^[3] For n \geq 2, SO(n) coincides with the connected component of the identity element in O(n), capturing all orientation-preserving orthogonal transformations.^[3] As a matrix Lie group, SO(n) inherits a smooth manifold structure from O(n), with the same Lie algebra \mathfrak{so}(n) consisting of skew-symmetric matrices.^[25] The dimension of this Lie group is n(n-1)/2, reflecting the degrees of freedom in specifying rotations while preserving orthogonality and determinant 1.^[26] A prominent example is SO(3), which parameterizes all proper rotations in three-dimensional Euclidean space. Elements of SO(3) can be represented via the axis-angle parameterization, where each rotation is specified by a unit vector \mathbf{u} \in S^2 (the rotation axis) and an angle \theta \in [0, \pi] (the rotation amount), according to Euler's rotation theorem.^[27] Alternatively, Euler angles provide a parameterization using three angles (\alpha, \beta, \gamma), corresponding to successive rotations about coordinate axes (e.g., Z-Y-X convention: rotation by \gamma about z, then \beta about y, then \alpha about x).^[28] SO(n) is doubly covered by the spin group Spin(n), a simply connected Lie group that projects onto SO(n) via a 2-to-1 homomorphism with kernel \{\pm I\}.^[29]

Reflections and canonical forms

Householder reflections provide a fundamental means of constructing elements of the orthogonal group O(n). A Householder reflection is an orthogonal transformation defined by a unit vector u \in \mathbb{R}^n with \|u\| = 1, given by the matrix

H = I - 2uu^T,

where I is the n \times n identity matrix and u^T denotes the transpose of u. This matrix represents a reflection across the hyperplane orthogonal to u, fixing all vectors in that hyperplane while mapping u to -u.^[30] The eigenvalues of a Householder reflection matrix H consist of $1 with multiplicity n-1 and -1 with multiplicity $1, reflecting its geometric action: it leaves an (n-1)-dimensional subspace unchanged (eigenvalue $1) and reverses the direction along the normal vector (eigenvalue -1). This spectral structure underscores the reflection's determinant of -1 and its role as an improper orthogonal transformation.^[31] The full orthogonal group O(n) over the reals, for n \geq 2, is generated by such reflections. Any orthogonal matrix can be expressed as a product of Householder reflections, a property rooted in the fact that reflections form a generating set for the group, allowing decomposition into basic geometric operations. This generation extends to the infinite Coxeter group structure associated with the root system of type B_n/C_n, where reflections correspond to the Weyl group generators.^[32] A key application of Householder reflections lies in the canonical forms of quadratic forms under orthogonal transformations, as described by Sylvester's law of inertia. For a real symmetric bilinear form defined by a matrix A, there exists an orthogonal matrix Q \in O(n) such that Q^T A Q = D, where D is a diagonal matrix with entries +1 (say, p times), -1 (say, q times), and $0 (say, r times), with p + q + r = n. The triple (p, q, r) is the inertia of the form and remains invariant under all orthogonal changes of basis, classifying quadratic forms up to orthogonal equivalence. This diagonalization highlights the orthogonal group's ability to preserve the signature while simplifying the form.^[33] In numerical algorithms, Householder reflections enable the QR decomposition of a full-rank matrix A \in \mathbb{R}^{m \times n} (with m \geq n) as A = QR, where Q \in O(m) (or more precisely, in the Stiefel manifold) and R is upper triangular. The process applies a sequence of n Householder transformations H_k to introduce zeros below the diagonal in successive columns of A, yielding Q = H_1 H_2 \cdots H_n and R = H_n \cdots H_2 H_1 A. Each H_k is chosen to reflect the k-th column subvector onto a multiple of the standard basis vector, ensuring numerical stability and efficiency in applications like solving least squares problems.^[34]

Symmetry of spheres

The orthogonal group O(n) acts on the unit sphere S^{n-1} = \{ x \in \mathbb{R}^n \mid \|x\| = 1 \} in \mathbb{R}^n via the natural linear action Q \cdot x for Q \in O(n) and x \in S^{n-1}. This action preserves the sphere because orthogonal matrices satisfy Q^T Q = I, implying \|Q x\| = \|x\| for all x, and thus map unit vectors to unit vectors.^[35] The special orthogonal group SO(n), consisting of the determinant-1 elements of O(n), forms the connected component of rotations that act as orientation-preserving isometries on S^{n-1}. This subgroup acts transitively on S^{n-1} for n \geq 2, meaning that for any two points p, q \in S^{n-1}, there exists R \in SO(n) such that R p = q. The stabilizer of a fixed point, such as the standard basis vector e_1 = (1, 0, \dots, 0), is isomorphic to SO(n-1), which acts on the orthogonal hyperplane \{e_1\}^\perp. Consequently, S^{n-1} is diffeomorphic to the homogeneous space SO(n)/SO(n-1).^[35]^[36] The full group O(n) extends this action to include orientation-reversing isometries, such as reflections. In particular, the central inversion -I \in O(n) realizes the antipodal map x \mapsto -x, which preserves the sphere and pairs antipodal points. While the isometry group of the round S^{n-1} embedded in \mathbb{R}^n is precisely O(n), the focus here is on this linear orthogonal action rather than the broader isometry group of S^n in \mathbb{R}^{n+1}, which is O(n+1).^[3]^[35] A concrete example arises for n=3, where SO(3) parametrizes the rotations of the 2-sphere S^2, corresponding to the symmetry group of a sphere in three-dimensional space, such as in rigid body dynamics or spherical geometry. Any orientation-preserving isometry of S^2 is a rotation around some axis through the origin.^[37]

Algebraic structure

Matrix groups over fields

The orthogonal group O(n, F) over a field F of characteristic not 2 is defined as the subgroup of the general linear group GL(n, F) consisting of all n \times n matrices A with entries in F such that A^T A = I_n, where A^T is the transpose of A and I_n is the n \times n identity matrix. This condition ensures that A preserves the standard bilinear form B(x, y) = x^T y on the vector space F^n, making O(n, F) the group of linear isometries with respect to this form. The assumption on the characteristic avoids complications with the definition of the transpose and symmetric forms, as in characteristic 2 the theory requires separate treatment using alternating forms.^[38] Explicit examples of elements in O(n, F) can be constructed over specific fields like the rationals \mathbb{Q} or finite fields \mathbb{F}_p with p odd. Over \mathbb{Q}, a simple rotation matrix in O(2, \mathbb{Q}) is given by

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},

which has order 4 and satisfies the defining relation.^[3] Over \mathbb{F}_p for odd prime p, diagonal matrices with entries \pm 1 on the diagonal form reflections or sign changes that belong to the group, such as the matrix with a single -1 and the rest 1's, preserving the form.^[39] These examples illustrate how the group structure manifests in low dimensions over fields with explicit arithmetic. The orthogonal group acts on the space of quadratic forms over F via the adjoint action: for a quadratic form q(v) = v^T S v associated to a symmetric matrix S, the action of A \in O(n, F) is A \cdot q (v) = q(A^T v) = (A^T v)^T S (A^T v) = v^T A S A^T v, yielding the transformed form with matrix A S A^T.^[40] Since A^{-1} = A^T, this action preserves the standard form q(v) = v^T v (where S = I_n), and more generally, it acts on the set of non-degenerate quadratic forms by congruence, classifying them up to isomorphism under the field's properties.^[41] Elements of finite order in O(n, F) generate cyclic subgroups, which are crucial for understanding the group's torsion. For instance, over fields containing roots of unity, rotations by angles corresponding to roots of cyclotomic polynomials yield elements of order dividing the field's multiplicative order; over \mathbb{Q}, such elements are limited to orders 1, 2, 3, 4, and 6 due to the possible rotation angles in rational matrices.^[42] These cyclic subgroups embed into tori within the group, but their explicit matrix forms highlight the discrete symmetries preserved by the orthogonal condition. Unlike the unitary group over the complex numbers \mathbb{C}, where matrices satisfy A^* A = I_n with A^* the conjugate transpose, the orthogonal group over \mathbb{C} uses only the transpose, ignoring complex conjugation.^[43] This distinction means O(n, \mathbb{C}) is not a subgroup of the unitary group U(n), as orthogonal matrices need not preserve the Hermitian form, leading to different topological and algebraic behaviors over \mathbb{C}.^[44]

Algebraic groups

Over an algebraically closed field k of characteristic not equal to 2, the orthogonal group O(n,k) is defined as the closed subgroup of the general linear group GL(n,k) consisting of matrices A satisfying A^T A = I_n, where A^T denotes the transpose of A and I_n is the n \times n identity matrix. This set of equations defines a smooth affine variety, thereby endowing O(n,k) with the structure of a linear algebraic group. The special orthogonal group SO(n,k) is the kernel of the determinant homomorphism O(n,k) \to \{\pm 1\}, which identifies it as the connected component of the identity in O(n,k).^[45] Both O(n,k) and SO(n,k) are reductive linear algebraic groups, meaning their unipotent radicals are trivial and they admit a Levi decomposition for every parabolic subgroup.^[45] For n \geq 3, SO(n,k) is in fact semisimple, with finite center and no nontrivial abelian normal subgroups.^[45] Reductive groups like these possess a well-developed theory of subgroups, including Borel subgroups, which are maximal connected solvable subgroups, and parabolic subgroups, which contain a Borel subgroup and admit a semidirect product decomposition into a Levi factor and a unipotent radical.^[46] In the case of the orthogonal group, a standard Borel subgroup stabilizes a complete flag of subspaces isotropic with respect to the underlying nondegenerate symmetric bilinear form, while parabolic subgroups correspond to partial such flags and preserve the induced form on quotients.^[46] The classification of these groups as semisimple algebraic groups relies on their associated root systems and corresponding Dynkin diagrams. For the odd-dimensional special orthogonal group SO(2m+1,k), the root system is of type B_m, with Dynkin diagram comprising m nodes connected by m-1 edges: the first m-1 edges are single bonds, and the final edge is a double bond with an arrow indicating the shorter root.^[47] For the even-dimensional case SO(2m,k), the root system is of type D_m, whose Dynkin diagram features m nodes with single bonds connecting the first m-2 pairs, followed by the (m-1)-th node linked by single bonds to two terminal nodes.^[47] These diagrams encode the relations among simple roots and underpin the structure theory, including the Weyl group and representation theory of the groups.^[47] Over the complex numbers \mathbb{C}, an algebraically closed field, the orthogonal group is split reductive, admitting a maximal torus that splits completely over \mathbb{C} (isomorphic to (\mathbb{G}_m)^r where r is the rank).^[48] In contrast, over the reals \mathbb{R}, the standard orthogonal group O(n,\mathbb{R}) preserving the positive definite quadratic form \sum x_i^2 is a compact real form of the complex group and lacks a split maximal torus, corresponding to a non-split real structure.^[48] Split real forms of the orthogonal group arise instead from indefinite quadratic forms, such as O(p,q,\mathbb{R}) with p+q=n and \min(p,q) \geq 1, where a split torus exists when the signature allows a suitable choice of Cartan subalgebra.^[48]

Maximal tori and Weyl groups

In the special orthogonal group \mathrm{SO}(2k), a maximal torus T consists of block-diagonal matrices formed by k copies of $2 \times 2 rotation matrices \begin{pmatrix} \cos \theta_i & -\sin \theta_i \\ \sin \theta_i & \cos \theta_i \end{pmatrix} for \theta_i \in [0, 2\pi), embedded along the diagonal. This subgroup is maximal abelian and connected, isomorphic to the k-torus (S^1)^k. Similarly, in \mathrm{SO}(2k+1), a maximal torus is obtained by adjoining a $1 \times 1 identity block to the structure in \mathrm{SO}(2k), yielding an embedding of (S^1)^k into the odd-dimensional group. The Weyl group W of the orthogonal group is defined as the quotient N_G(T)/T, where N_G(T) is the normalizer of T in G = \mathrm{SO}(n). For G = \mathrm{O}(n) or \mathrm{SO}(2k+1), W is isomorphic to the hyperoctahedral group of rank k, which consists of all signed permutations of k elements and has order $2^k k!; this group can be realized as the subgroup of O(k) generated by permutations and diagonal sign changes.^[49] For G = \mathrm{SO}(2k) with k \geq 2, the Weyl group is the index-2 subgroup of the hyperoctahedral group comprising signed permutations with an even number of sign changes.^[49] The Weyl group acts on the maximal torus T by conjugation: elements of N_G(T) conjugate elements of T, and since T is abelian, this descends to a well-defined action of W on T. Identifying T with (S^1)^k via the angles \theta = (\theta_1, \dots, \theta_k), the action corresponds to permuting the coordinates and replacing \theta_i with -\theta_i (reflecting across the corresponding root hyperplane).^[50] This action is faithful and reflects the symmetry of the root system. The root system associated to \mathrm{SO}(2k+1) is of type B_k, with simple roots e_1 - e_2, \dots, e_{k-1} - e_k, e_k in the standard Euclidean space \mathbb{R}^k (where e_i are the basis vectors); the full roots are \pm e_i \pm e_j (i < j) and \pm e_i.^[49] For \mathrm{SO}(2k), the root system is of type D_k, with simple roots e_1 - e_2, \dots, e_{k-1} - e_k, e_{k-1} + e_k; the roots are \pm e_i \pm e_j (i < j) excluding the short roots.^[49] The Weyl group is generated by reflections across the hyperplanes orthogonal to these roots. The length function \ell(w) on the Weyl group counts the minimal number of simple reflections needed to express w, equivalent to the number of inversions in the permutation part plus the number of negative signs for type B_k. The Bruhat order on W is the partial order generated by covering relations w < ws_\alpha where \ell(ws_\alpha) = \ell(w) + 1 for simple roots \alpha, providing a combinatorial structure for decomposing elements and cells in the flag variety.^[51]

Topology

Low-dimensional cases

The orthogonal group O(1) over the reals consists of the $1 \times 1 orthogonal matrices, which are precisely the elements \pm 1, forming a discrete group isomorphic to \mathbb{Z}/2\mathbb{Z}.^[9] This group is trivial in the sense that it has only two elements and acts on the real line by preservation of the absolute value.^[9] In dimension 2, the orthogonal group O(2) is the group of isometries of the Euclidean plane preserving orientation up to reflection, and it is isomorphic to the infinite dihedral group D_\infty.^[52] This structure arises as the semidirect product \mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}, where the infinite cyclic subgroup corresponds to rotations and the \mathbb{Z}/2\mathbb{Z} factor to reflections.^[52] The special orthogonal subgroup SO(2) consists solely of rotations and is isomorphic to the circle group U(1), the multiplicative group of complex numbers of modulus 1, or topologically to the 1-sphere S^1./01%3A_Chapters/12%3A_The_Circle_Group) Explicitly, the isomorphism maps e^{i\theta} to the rotation matrix \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}./01%3A_Chapters/12%3A_The_Circle_Group) For dimension 3, the special orthogonal group SO(3) is the group of rotations of \mathbb{R}^3, which admits a double cover by the special unitary group SU(2), the group of $2 \times 2 unitary matrices with determinant 1.^[53] This covering map SU(2) \to SO(3) is a 2-to-1 homomorphism with kernel \{\pm I\}, reflecting that rotations by an angle \theta and $2\pi - \theta are identified in SO(3) but distinct in SU(2).^[54] Topologically, SO(3) is homeomorphic to the real projective space \mathbb{RP}^3, obtained as the quotient of the 3-sphere S^3 by the antipodal map, and its fundamental group is \mathbb{Z}/2\mathbb{Z}, indicating it is not simply connected.^[53]^[55] Furthermore, SO(3) is isomorphic to the projective special unitary group PSU(2) = SU(2)/\{\pm I\}, providing an identification with projective transformations preserving the Hermitian form on \mathbb{C}^2.^[56] The finite subgroups of SO(3) include the alternating group A_5 (icosahedral rotations), whose double cover in SU(2) is the binary icosahedral group of order 120, yielding quotients like the Poincaré homology sphere S^3 / binary icosahedral.^[57]

Fundamental group

The special orthogonal group SO(n) is path-connected for all n \geq 1. Its fundamental group is \pi_1(SO(2)) \cong \mathbb{Z} since SO(2) is diffeomorphic to the circle S^1, and \pi_1(SO(n)) \cong \mathbb{Z}/2\mathbb{Z} for n \geq 3.^[58] The orthogonal group O(n) consists of two path-connected components for n \geq 1, corresponding to matrices with determinant +1 (namely SO(n)) and determinant -1. These components are diffeomorphic via multiplication by a fixed reflection matrix, so \pi_1(O(n)) \cong \pi_1(SO(n)) in each case. For n=1, both SO(1) and O(1) are discrete (the trivial group and \{\pm 1\}, respectively), yielding trivial fundamental groups.^[59] To establish \pi_1(O(n)) \cong \pi_1(SO(n)), consider the determinant fibration \det: O(n) \to \{\pm 1\} with fiber SO(n). The long exact sequence of homotopy groups for this fibration is

\cdots \to \pi_1(SO(n)) \to \pi_1(O(n)) \to \pi_1(\{\pm 1\}) = \{0\} \to \pi_0(SO(n)) = \{0\} \to \pi_0(O(n)) \to \pi_0(\{\pm 1\}) = \mathbb{Z}/2\mathbb{Z} \to \{0\}.

The relevant portion for \pi_1 shows that the induced map \pi_1(SO(n)) \to \pi_1(O(n)) is an isomorphism, while the portion for \pi_0 confirms the two components of O(n).^[60]^[59] The group \mathbb{Z}/2\mathbb{Z} for n \geq 3 arises as the deck transformation group of the universal covering space of SO(n), given by the spin group Spin(n) \to SO(n), which is a double cover (i.e., 2-to-1). For n=2, the analogous covering S^1 \to SO(2) given by \theta \mapsto 2\theta is infinite, consistent with \pi_1(SO(2)) \cong \mathbb{Z}. For n=4, SO(4) is diffeomorphic to (S^3 \times S^3)/\mathbb{Z}_2, but the fundamental group remains \mathbb{Z}/2\mathbb{Z} as the quotient does not alter the 1-dimensional homotopy.^[59] In low dimensions, explicit computations align with these general results: for instance, \pi_1(SO(3)) \cong \mathbb{Z}/2\mathbb{Z} since SO(3) \cong \mathbb{RP}^3.

Homotopy groups

The homotopy groups of the orthogonal group O(n) and its connected component SO(n) exhibit rich structure, particularly in the stable regime where n is sufficiently large compared to the degree. The Bott periodicity theorem provides a complete description of the stable homotopy groups of the infinite orthogonal group O = \lim_{n \to \infty} O(n), asserting that \pi_k(O) \cong \pi_{k+8}(O) for all k \geq 0. Explicitly, these groups are \pi_0(O) = \mathbb{Z}/2\mathbb{Z}, \pi_1(O) = \mathbb{Z}/2\mathbb{Z}, \pi_2(O) = 0, \pi_3(O) = \mathbb{Z}, \pi_4(O) = 0, \pi_5(O) = 0, \pi_6(O) = 0, \pi_7(O) = \mathbb{Z}, and then the pattern repeats every 8 dimensions. For the special orthogonal group, the stable homotopy groups coincide with those of O in positive degrees, since \pi_k(SO) \cong \pi_k(O) for k > 0, reflecting the double cover SO \to O after the basepoint component. The stable homotopy groups of O are intimately connected to real K-theory, or KO-theory. Specifically, the reduced KO-theory groups \tilde{KO}(S^m) classify stable real vector bundles over the m-sphere up to isomorphism, and by the clutching construction, these are isomorphic to \pi_{m-1}(O). Thus, \pi_k(O) \cong \tilde{KO}(S^{k+1}), linking the topological structure of O to the coefficients of the KO-theory spectrum.^[61] This identification underscores the role of Bott periodicity in both algebraic topology and index theory, where the 8-fold periodicity governs the structure of real vector bundles and elliptic operators. For finite n, the unstable homotopy groups \pi_k(O(n)) and \pi_k(SO(n)) can be computed recursively using fibration sequences arising from the inclusions of orthogonal groups. The principal fibration O(n) \to O(n+1) \to S^n induces a long exact sequence in homotopy: \cdots \to \pi_{k+1}(S^n) \to \pi_k(O(n)) \to \pi_k(O(n+1)) \to \pi_k(S^n) \to \cdots. In the range $0 < k < n-1, the sphere groups vanish, yielding isomorphisms \pi_k(O(n)) \cong \pi_k(O(n+1)), which stabilize to the Bott groups as n \to \infty. A similar fibration holds for SO(n), with fiber S^{n-1}. These sequences allow explicit determination of unstable groups by inducting from low dimensions and incorporating the stable values. Geometrically, the homotopy groups \pi_{n+k-1}(SO(n)) admit interpretations in terms of framed cobordisms and immersions. By the Pontryagin-Thom construction and the Smale-Hirsch theorem, elements correspond to framed cobordism classes of (k-1)-dimensional manifolds immersed (or embedded, in stable ranges) into \mathbb{R}^n, where the framing provides a trivialization of the normal bundle. In particular, generators often arise from standard immersions like the Veronese embedding or Haefliger links, classifying oriented manifolds up to framed bordism in the metastable range k < \frac{3}{2}(n-1). The successive connected covers in the Whitehead tower of O(n) systematically kill the low-degree homotopy groups, aligning with the stages of Bott periodicity. Starting from O(n), the 0-connected cover is SO(n); the 3-connected cover is the spin group \operatorname{Spin}(n); the 7-connected cover is the string group \operatorname{String}(n); and the 15-connected cover is the fivebrane group \operatorname{Fivebrane}(n). These structures encode higher refinements of orthogonal frames, such as spin structures for fermions or string structures to cancel anomalies in topological terms.^[62]

Orthogonal groups over other fields

Indefinite quadratic forms over reals

The indefinite orthogonal group O(p,q) consists of all real (p+q) \times (p+q) matrices A such that A^T \eta A = \eta, where \eta is the diagonal matrix \operatorname{diag}(I_p, -I_q) with I_k the k \times k identity matrix, corresponding to the preservation of a quadratic form on \mathbb{R}^{p+q} with signature (p,q) (that is, p positive and q negative eigenvalues in the diagonalized form).^[63] This quadratic form can be expressed as Q(x) = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_{p+q}^2 for x = (x_1, \dots, x_{p+q}) \in \mathbb{R}^{p+q}.^[4] Assuming p \geq 1 and q \geq 1, the group O(p,q) has four connected components, determined by the sign of the determinant (which is \pm 1) and whether it preserves or reverses the temporal orientation (orthochronous versus antichronous).^[64] The identity component O^+(p,q) consists of those matrices with \det A = 1 and preserving the forward light cone, while the full special orthogonal group SO(p,q) has two components.^[65] A maximal compact subgroup of O(p,q) is O(p) \times O(q), which acts on the positive and negative definite subspaces respectively and is unique up to conjugation.^[66] This subgroup is compact because both O(p) and O(q) are compact Lie groups, and it achieves maximality as no larger compact subgroup can embed without violating the non-compact nature of the overall group.^[67] The Iwasawa decomposition of O(p,q) (or more precisely, its identity component) expresses elements as k a n where K = O(p) \times O(q) is the maximal compact, A is the vector subgroup of diagonal matrices with positive entries exponentiating the Cartan subalgebra, and N is the unipotent radical consisting of upper triangular matrices with 1s on the diagonal.^[68] This decomposition is analytic and unique, facilitating harmonic analysis and representation theory on the group.^[69] A prominent example is O(1,3), the Lorentz group, which preserves the Minkowski quadratic form Q(t,x,y,z) = -t^2 + x^2 + y^2 + z^2 on \mathbb{R}^{1,3}, central to the mathematical structure of special relativity where it acts as the symmetry group of spacetime intervals. In this case, the four components correspond to proper/improper orthochronous/anti-chronous transformations, with the proper orthochronous subgroup SO^+(1,3) being the connected component of the identity.^[70]

Complex orthogonal groups

The complex orthogonal group O(n, \mathbb{C}) consists of all n \times n complex matrices A such that A^T A = I_n, where A^T denotes the transpose of A and I_n is the identity matrix. This definition arises from the requirement that elements preserve the standard symmetric bilinear form \langle x, y \rangle = x^T y on \mathbb{C}^n. As a subgroup of the general linear group \mathrm{GL}(n, \mathbb{C}), O(n, \mathbb{C}) is a complex algebraic group and a complex Lie group of dimension n(n-1)/2. The special complex orthogonal group \mathrm{SO}(n, \mathbb{C}) is the normal subgroup of index 2 comprising those elements with determinant 1, defined as \{ A \in \mathrm{SL}(n, \mathbb{C}) \mid A^T A = I_n \}. Unlike the real orthogonal group, which is compact, O(n, \mathbb{C}) is non-compact as a real manifold of dimension n(n-1), reflecting its complex structure. Algebraically, O(n, \mathbb{C}) serves as the complexification of the real orthogonal group O(n, \mathbb{R}), obtained by extending scalars from \mathbb{R} to \mathbb{C}; this isomorphism holds in the category of algebraic groups. For n odd, O(n, \mathbb{C}) is of type B_{(n-1)/2}, and for n even, it relates to type D_{n/2}, with the special subgroup capturing the semisimple structure. Topologically, O(n, \mathbb{C}) has two connected components, distinguished by the sign of the determinant, with \mathrm{SO}(n, \mathbb{C}) forming the identity component. The fundamental group \pi_1(O(n, \mathbb{C})) is \mathbb{Z}/2\mathbb{Z} for n \geq 2, mirroring the real case but computed via the complex manifold structure; higher homotopy groups \pi_k(O(n, \mathbb{C})) for k \geq 2 coincide with those of O(n, \mathbb{R}) due to the stable homotopy equivalence in the complex setting. The complex orthogonal group relates to the complex symplectic group \mathrm{Sp}(2m, \mathbb{C}) via exterior forms: the alternating form preserved by the symplectic group can be constructed from the wedge product of the symmetric form, linking representations in the exterior algebra. Over \mathbb{C}, which is algebraically closed, all non-degenerate quadratic forms on an n-dimensional space are congruent to the standard sum of squares \sum_{i=1}^n x_i^2. Consequently, there are no distinct indefinite complex orthogonal groups analogous to the real indefinite case O(p,q); all such groups are isomorphic to the standard O(n, \mathbb{C}).

Orthogonal groups over finite fields

Over finite fields \mathbb{F}_q with q elements, where q is a power of a prime, the orthogonal group O(n,q) is defined as the group of n \times n matrices over \mathbb{F}_q that preserve a non-degenerate quadratic form Q on the vector space \mathbb{F}_q^n, i.e., matrices A such that Q(Av) = Q(v) for all v \in \mathbb{F}_q^n. The classification of these groups depends on the characteristic of the field: when \mathrm{char}(\mathbb{F}_q) \neq 2, the theory aligns closely with the real case via Witt decomposition, while in characteristic 2, quadratic forms behave differently due to the coincidence of symmetric and alternating bilinear forms.^[16] When \mathrm{char}(\mathbb{F}_q) \neq 2, non-degenerate quadratic forms on \mathbb{F}_q^n are classified up to isometry by their dimension and type, determined by the Witt index (the dimension of a maximal isotropic subspace). For odd dimension n = 2m+1, there is essentially one isometry class, denoted O(2m+1,q), with Witt index m. For even dimension n = 2m, there are two distinct classes: the plus type O^+(2m,q) with Witt index m (hyperbolic form), and the minus type O^-(2m,q) with Witt index m-1 (elliptic form). These types correspond to the Witt groups of quadratic forms over \mathbb{F}_q. The plus and minus types are distinguished by the Dickson invariant, a group homomorphism d: O(2m,q) \to \mathbb{Z}/2\mathbb{Z} defined via the determinant of the restriction to a hyperbolic plane or, equivalently, by the action on the Clifford algebra; it takes value 0 on O^+(2m,q) and 1 on O^-(2m,q).^[39] The orders of these groups are given by explicit formulas derived from counting isometries via recursive decomposition of the form. For the odd-dimensional case, |O(2m+1,q)| = 2 q^m \prod_{i=1}^m (q^{2i} - 1). For even dimension, |O^+(2m,q)| = 2 (q^m - 1) q^{m(m-1)/2} \prod_{i=1}^{m-1} (q^{2i} - 1) and |O^-(2m,q)| = 2 (q^m + 1) q^{m(m-1)/2} \prod_{i=1}^{m-1} (q^{2i} - 1). These formulas reflect the structure: the factor of 2 accounts for determinant \pm 1, the q^{m(m-1)/2} term arises from the unipotent radical of a Borel subgroup, and the products count the Weyl group contributions. The special orthogonal subgroups SO(n,q) have index 2 in O(n,q) except in small cases like O^+(2,q) \cong S_3. Orthogonal groups over \mathbb{F}_q with \mathrm{char} \neq 2 are generated by reflections and act irreducibly on the natural module \mathbb{F}_q^n, with representations classified via Brauer characters or Deligne-Lusztig theory for simple quotients like \Omega^\pm(2m,q).^[16]^[39] In characteristic 2, where q = 2^e for e \geq 1, the situation differs because every symmetric bilinear form is alternating (its diagonal vanishes), so quadratic forms Q are defined independently via Q(av) = a^2 Q(v) and the associated bilinear form B(u,v) = Q(u+v) - Q(u) - Q(v), which is alternating and non-degenerate for non-singular Q. Orthogonal groups O(n,2^e) preserve such quadratic forms, but classification relies on the theory of quadratic forms over fields of char 2 rather than just bilinear forms. For odd dimension n = 2m+1, there is a unique isometry class of non-singular quadratic forms (up to scaling), yielding O(2m+1,2^e) with Witt index m. For even dimension n = 2m, there are two classes: hyperbolic (Witt index m) and elliptic (Witt index m-1), often denoted without ^\pm but distinguished by the Arf invariant (a \mathbb{Z}/2\mathbb{Z}-valued functional on the space of quadratic forms modulo hyperbolic planes). In char 2, orthogonal groups must be carefully distinguished from unitary groups over \mathbb{F}_{q^2}, as the latter preserve sesquilinear forms; however, certain quadratic forms in char 2 induce unitary structures upon extension. The Dickson invariant still applies in even dimension to separate the types, but computations involve the quotient by the kernel of the squaring map. Orders follow similar recursive formulas adjusted for char 2, such as |O(2m+1,2^e)| = 2 \cdot 2^{m e} \prod_{i=1}^m (2^{2 e i} - 1), with analogous expressions for even cases involving factors like $2^{m e} \pm 2^{m e / 2}. These groups are generated by transvections and act on the natural module, with irreducible representations over \mathbb{F}_{2^e} studied via modular representation theory.^[16]^[39]

Advanced topics

Spinor norm

The spinor norm is a canonical group homomorphism N: O(V) \to k^\times / (k^\times)^2 defined for the orthogonal group O(V) of a non-degenerate quadratic space (V, q) over a field k of characteristic not 2. This map arises in the arithmetic theory of quadratic forms and captures an invariant of orthogonal transformations modulo squares in k. It plays a key role in distinguishing elements within the special orthogonal group SO(V), where the kernel of N restricted to SO(V) often coincides with the commutator subgroup [SO(V), SO(V)].^[71] The spinor norm factors through the determinant homomorphism \det: O(V) \to \{ \pm 1 \} (identified with a subgroup of k^\times / (k^\times)^2) and the Dickson invariant d: O(V) \to \{ \pm 1 \} (identified with a subgroup of k^\times / (k^\times)^2), which is determined by the dimension of V and the discriminant of q. The Dickson invariant is a group homomorphism related to the determinant and the parity of the dimension. This decomposition aids in classifying orthogonal elements and understanding the structure of O(V).^[72] Explicitly, the spinor norm can be computed via decompositions into reflections. For a reflection \sigma_v across the hyperplane orthogonal to a non-isotropic vector v \in V, N(\sigma_v) = q(v) (k^\times)^2. Since every element of O(V) is a product of an even or odd number of such reflections (even for SO(V)), the spinor norm is multiplicative: if g = \sigma_{v_1} \cdots \sigma_{v_r}, then N(g) = \prod_{i=1}^r q(v_i) \ (k^\times)^2. This allows practical computation for products of reflections, independent of the choice up to squares.^[72] In the context of quadratic forms over number fields, the spinor norm is instrumental in the Hasse principle, particularly for determining when a quadratic form represents another locally everywhere. It governs the spinor genus of integral quadratic lattices, where two lattices in the same genus are in the same spinor genus if their orthogonal groups share the same spinor norms at all places; this refines the local-global principle by accounting for global obstructions via idele class group mappings. The principle holds for representations by ternary and higher-dimensional forms under spinor norm compatibility. The spinor norm is intimately related to the Clifford algebra C(V, q) and the spin group \Spin(V, q), the kernel of the reduced norm \Nrd: C(V, q)^\times \to k^\times restricted to the even part, which double covers SO(V). The spinor norm on SO(V) is induced by the Clifford norm on the preimage in \Spin(V, q), well-defined modulo squares since elements in the center \{ \pm 1 \} have norm 1. This connection embeds the spinor norm in the broader framework of algebraic groups and automorphic forms.^[73]

Galois cohomology

The Galois cohomology group H^1(k, O(V)), where k is a field, V is the standard n-dimensional vector space over the separable closure k^s equipped with the split quadratic form q_0, and O(V) is its orthogonal group, classifies isomorphism classes of quadratic forms of dimension n over k. Specifically, elements of this pointed set correspond to O(V)-torsors over k, which via Galois descent are precisely the quadratic spaces (W, q) over k isomorphic to (V, q_0) over k^s. The trivial torsor corresponds to the split form q_0, and two quadratic forms are isomorphic over k if and only if their associated cocycles are cohomologous. A key invariant arising from this classification is the Clifford invariant, which links quadratic forms to the Brauer group \Br(k). The even Clifford algebra C_0(q) of a quadratic form q is a central simple algebra over k of degree $2^{m-1} for \dim q = 2m or $2^m for \dim q = 2m+1, and its class [C_0(q)] \in {}_2\Br(k) (the 2-torsion) provides a cohomological obstruction to isotropy. Merkurjev's theorem establishes that every element of {}_2\Br(k) is represented by such a Clifford algebra for some quadratic form, yielding a surjection from the Witt group of quadratic forms to {}_2\Br(k). This connection extends to central simple algebras with orthogonal involutions, where the Galois cohomology of the associated special orthogonal group encodes similarity classes.^[74] Over number fields, local-global principles for quadratic forms are governed by Hasse invariants, which are local cohomological data in H^1(k_v, O(V)) for completions k_v at places v. The Hasse-Minkowski theorem asserts that a quadratic form over a number field k is isotropic if and only if it is isotropic over every k_v, with the global isomorphism class determined by matching local Hasse symbols (products of Hilbert symbols) across all places, up to the product-one relation from class field theory. Failures of stronger principles, such as the Hasse principle for rational points on orthogonal varieties, can be explained by Brauer-Manin obstructions in higher cohomology, but for the classification itself, the local data suffice.^[75] Torsors under O(V) can be realized geometrically as affine varieties over k equipped with a quadratic form q such that the associated projective quadric is a principal homogeneous space for the projective orthogonal group PO(V). These torsors are trivialized over splitting fields of q, and over number fields, their arithmetic is studied via descent from local models; for instance, non-split torsors correspond to anisotropic forms like the norm form of a quaternion algebra, linking back to the Brauer group.^[76]

Lie algebra

The Lie algebra \mathfrak{so}(n) of the real orthogonal group O(n) (or its connected component SO(n)) consists of all n \times n skew-symmetric real matrices, i.e., \mathfrak{so}(n) = \{ X \in \mathfrak{gl}(n, \mathbb{R}) \mid X^T = -X \}, equipped with the Lie bracket [X, Y] = XY - YX. This Lie algebra has dimension n(n-1)/2, as it is spanned by the basis elements E_{ij} - E_{ji} for $1 \leq i < j \leq n, where E_{kl} denotes the matrix with a 1 in the (k,l)-entry and zeros elsewhere.^[77]^[78] The complexification \mathfrak{so}(n, \mathbb{C}) is a semisimple Lie algebra (simple for n \neq 4) with a Cartan subalgebra \mathfrak{h} formed by diagonal matrices in a suitable basis. Its root system \Phi relative to \mathfrak{h} depends on the parity of n: if n = 2l (type D_l), then \Phi = \{ \pm e_i \pm e_j \mid 1 \leq i < j \leq l \}, where \{e_1, \dots, e_l\} is the standard basis of \mathbb{R}^l; if n = 2l+1 (type B_l), then \Phi = \{ \pm e_i \pm e_j \mid 1 \leq i < j \leq l \} \cup \{ \pm e_i \mid 1 \leq i \leq l \}. The root spaces \mathfrak{g}_\alpha are one-dimensional for each root \alpha \in \Phi, and the algebra decomposes as \mathfrak{so}(n, \mathbb{C}) = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha. A Chevalley basis for \mathfrak{so}(n, \mathbb{C}) consists of the Cartan elements H_i (corresponding to simple roots), positive root vectors X_\alpha, and negative root vectors Y_\alpha = X_{-\alpha}, with structure constants that are integers, ensuring integrality properties useful for constructing Chevalley groups.^[79]^[80] The adjoint representation \mathrm{Ad}: \mathfrak{so}(n) \to \mathfrak{gl}(\mathfrak{so}(n)), given by \mathrm{Ad}_X(Y) = [X, Y], is faithful for n \geq 3, as \mathfrak{so}(n) is semisimple with trivial center in these cases (noting that \mathfrak{so}(4) \cong \mathfrak{so}(3) \oplus \mathfrak{so}(3) is semisimple but not simple). This representation embeds \mathfrak{so}(n) as a subalgebra of skew-symmetric matrices acting on itself.^[77]^[81] The Killing form B(X, Y) = \operatorname{Tr}(\mathrm{ad}_X \circ \mathrm{ad}_Y) on \mathfrak{so}(n, \mathbb{R}) is negative definite, which characterizes it as the compact real form of the complex orthogonal Lie algebra \mathfrak{so}(n, \mathbb{C}). Explicitly, B(X, Y) = (n-2) \operatorname{Tr}(XY) for X, Y \in \mathfrak{so}(n, \mathbb{R}), confirming non-degeneracy and negative definiteness. This property aligns with the compactness of SO(n) and facilitates the study of representations and invariant bilinear forms.^[82]^[83]

Lie subgroups and supergroups

The special orthogonal group \mathrm{SO}(n) is a connected Lie subgroup of index 2 in the orthogonal group \mathrm{O}(n), defined as the kernel of the determinant homomorphism \det: \mathrm{O}(n) \to \{\pm 1\}. It consists of all proper rotations preserving the standard Euclidean inner product on \mathbb{R}^n, and its Lie algebra is the space of skew-symmetric matrices \mathfrak{so}(n). As a simple Lie group for n \geq 3, \mathrm{SO}(n) plays a central role in the structure of \mathrm{O}(n), which is a semidirect product \mathrm{SO}(n) \rtimes \mathbb{Z}/2\mathbb{Z}.^[3] The orthogonal group \mathrm{O}(n-1) embeds as a closed Lie subgroup of \mathrm{O}(n) via the stabilizer of a unit vector, such as the first standard basis vector e_1 \in \mathbb{R}^n. This embedding identifies \mathrm{O}(n-1) with the block matrices of the form \begin{pmatrix} 1 & 0 \\ 0 & Q \end{pmatrix}, where Q \in \mathrm{O}(n-1), preserving the orthogonal structure on the orthogonal complement of \mathrm{span}\{e_1\}. Such stabilizers are maximal parabolic subgroups in the context of representation theory, and for n \geq 3, \mathrm{O}(n-1) is a maximal connected Lie subgroup of \mathrm{O}(n) up to conjugacy, except in dimensions 4 and 8 where exceptional isomorphisms occur.^[84] Finite Lie subgroups of \mathrm{O}(n) include the dihedral groups D_{2k}, which embed into \mathrm{O}(2) \subset \mathrm{O}(n) for n \geq 2 as the symmetries of a regular k-gon in the xy-plane. The dihedral group D_{2k} is generated by a rotation by $2\pi/k and a reflection, with presentation \langle r, s \mid r^k = s^2 = 1, s r s^{-1} = r^{-1} \rangle, and it acts faithfully on \mathbb{R}^2 while fixing the higher coordinates. These groups are examples of reflection groups and appear in classifications of finite subgroups of \mathrm{O}(n), alongside cyclic and polyhedral groups for higher dimensions.^[85] The conformal orthogonal group \mathrm{CO}(n), also denoted \mathrm{O}(n) \times \mathbb{R}^+, extends \mathrm{O}(n) by incorporating positive scalar multiplications, forming the semidirect product \mathbb{R}^+ \ltimes \mathrm{O}(n) where scalings act by conjugation. It consists of similitudes—linear transformations f: \mathbb{R}^n \to \mathbb{R}^n satisfying \langle f(x), f(y) \rangle = \lambda \langle x, y \rangle for some \lambda > 0—and serves as the structure group for conformal geometry on Riemannian manifolds. The full group including orientation-reversing elements is \mathbb{R}^* \ltimes \mathrm{O}(n), but the connected component is \mathrm{CO}^+(n) = \mathbb{R}^+ \ltimes \mathrm{SO}(n), with Lie algebra \mathfrak{co}(n) = \mathbb{R} \oplus \mathfrak{so}(n). This extension arises naturally in the study of Möbius transformations and angle-preserving maps.^[86] In supergeometry, the orthogonal supergroup \mathrm{O}(m|n) is a Lie supergroup preserving a consistent super quadratic form on the superspace \mathbb{R}^{m|n}, where the even (bosonic) part is the standard symmetric bilinear form on \mathbb{R}^m and the odd (fermionic) part involves an antisymmetric structure on \mathbb{R}^n. More commonly realized as the orthosymplectic supergroup \mathrm{OSp}(m|2n), it has even subgroup \mathrm{O}(m) \times \mathrm{Sp}(2n; \mathbb{R}) and odd part given by the orthosymplectic Lie superalgebra \mathfrak{osp}(m|2n), which includes matrices satisfying M^T J + J M = 0 for the super metric J = \begin{pmatrix} I_m & 0 \\ 0 & \Omega \end{pmatrix} with \Omega the symplectic form. This supergroup extends classical orthogonal groups to supersymmetric settings, appearing in representations of super Lie algebras and invariant theory for superalgebras, where the first fundamental theorem describes invariants under its action.^[87]^[88] The Pin and Spin groups provide double cover extensions of the orthogonal groups. The Spin group \mathrm{Spin}(n) is the unique simply connected double cover of \mathrm{SO}(n) for n \geq 3, with kernel \{\pm 1\}, constructed as the multiplicative group of even Clifford algebra elements of norm 1. Similarly, the Pin group \mathrm{Pin}(n) double covers \mathrm{O}(n), incorporating odd Clifford elements, and splits into \mathrm{Pin}^+(n) and \mathrm{Pin}^-(n) depending on the signature. These covers are essential for spinor representations and are Lie groups whose Lie algebras are \mathfrak{so}(n), facilitating the study of framings and Dirac operators.^[89] An example of embeddings involving the orthogonal group is the faithful representation of the general linear group \mathrm{GL}(n, \mathbb{R}) into the indefinite orthogonal group \mathrm{O}(n,n) \subset \mathrm{O}(2n), acting on \mathbb{R}^n \oplus (\mathbb{R}^n)^* by g \cdot (v, \phi) = (g v, \phi \circ g^{-1}), preserving the hyperbolic form \langle (v,\phi), (w,\psi) \rangle = \phi(w) + \psi(v). This realizes \mathrm{GL}(n) as a closed Lie subgroup of an orthogonal group in dimension $2n, highlighting connections between linear and orthogonal structures in higher dimensions.^[90]

Discrete subgroups

Discrete subgroups of orthogonal groups encompass both finite and infinite cases, playing key roles in geometry, crystallography, and number theory. Finite discrete subgroups are compact and arise naturally as symmetry groups of polyhedra or reflection arrangements, while infinite ones often exhibit lattice-like structures and act properly discontinuously on Euclidean spaces. Finite subgroups of the orthogonal group O(n) include the classical reflection groups, which are generated by reflections across hyperplanes and realized as Coxeter groups acting linearly on \mathbb{R}^n. These groups preserve a positive definite quadratic form and are finite precisely when the associated bilinear form is positive definite, as detailed in the classification of irreducible finite Coxeter groups corresponding to the Dynkin diagrams A_n, B_n, D_n, E_6, E_7, E_8, F_4, G_2, H_3, H_4, I_2(m). A prominent example in O(3) is the rotation groups of the Platonic solids: the tetrahedral group isomorphic to A_4 (order 12), the octahedral group to S_4 (order 24), and the icosahedral group to A_5 (order 60), all subgroups of SO(3); including reflections yields the full symmetry groups of orders 24, 48, and 120, respectively.^[91] The binary polyhedral groups provide finite subgroups of SU(2), the double cover of SO(3), which project onto the Platonic rotation groups via the canonical homomorphism SU(2) \to SO(3). These include the binary tetrahedral group (order 24), binary octahedral (order 48), and binary icosahedral (order 120), classified alongside cyclic and binary dihedral groups in the McKay correspondence linking them to ADE Dynkin diagrams.^[92] Infinite discrete subgroups include the crystallographic groups, which are discrete subgroups of the Euclidean isometry group E(n) = \mathbb{R}^n \rtimes O(n) acting cocompactly and freely on \mathbb{R}^n. The orthogonal part of such a group is a finite subgroup of O(n), while the translation subgroup is a full-rank lattice \mathbb{Z}^n. Bieberbach's theorems classify these: the translation subgroup has finite index, there are finitely many conjugacy classes up to dimension n, and isomorphism corresponds to conjugacy in the affine group.^[93] In three dimensions, there are 230 space groups, combining the 32 crystallographic point groups (finite subgroups of O(3)) with lattice translations.^[94] Arithmetic subgroups of SO(n, \mathbb{R}) form another important class of infinite discrete subgroups, obtained as integer points of the algebraic group SO(n) defined over \mathbb{Q}. For instance, SO(n, \mathbb{Z}) = SL(n, \mathbb{Z}) \cap SO(n, \mathbb{R}) consists of integer matrices with determinant 1 preserving the standard quadratic form, and it is an arithmetic lattice in SO(n, \mathbb{R}) for n \geq 3. These groups are commensurable with principal congruence subgroups and have finite covolume, with applications to hyperbolic geometry when the form is indefinite.^[95]

Covering and quotient groups

The spin group \operatorname{Spin}(n), for n \geq 3, is the universal covering group of the special orthogonal group \operatorname{SO}(n), providing a double cover with kernel isomorphic to \mathbb{Z}/2\mathbb{Z}.^[96] This covering map \operatorname{Spin}(n) \to \operatorname{SO}(n) arises from the construction via Clifford algebras, where \operatorname{Spin}(n) consists of even elements in the Clifford algebra that normalize the volume element, ensuring the kernel is \{ \pm 1 \}.^[97] The simply connected nature of \operatorname{Spin}(n) reflects the fundamental group \pi_1(\operatorname{SO}(n)) \cong \mathbb{Z}/2\mathbb{Z} for n \geq 3.^[71] Similarly, the pin group \operatorname{Pin}(n) serves as a double cover of the full orthogonal group \operatorname{O}(n), with kernel \mathbb{Z}/2\mathbb{Z}.^[98] There are two variants, \operatorname{Pin}^+(n) and \operatorname{Pin}^-(n), distinguished by the image of the reflections; the restriction of the covering map to the connected component \operatorname{SO}(n) yields the spin double cover. These covers are essential for defining spin structures on manifolds and handling orientation-reversing transformations in physics applications like the Dirac equation.^[98] A key quotient of the special orthogonal group is \operatorname{SO}(n)/\operatorname{SO}(n-1) \cong S^{n-1}, the (n-1)-dimensional sphere, arising from the transitive action of \operatorname{SO}(n) on S^{n-1} with stabilizer \operatorname{SO}(n-1) at a fixed point.^[99] This homogeneous space structure highlights the role of orthogonal groups in spherical geometry and symmetric spaces.^[100] In exceptional cases, such as dimension 8, the Weyl group of the E_8 root system provides a double cover of the orthogonal group O^+(8,2) over the finite field \mathbb{F}_2, illustrating a discrete analog of covering phenomena tied to the E_8 lattice.^[101] The E_8 lattice itself, as an even unimodular lattice in \mathbb{R}^8, admits an orthogonal group whose finite quotients encode symmetries beyond classical cases.^[102] Higher-dimensional covers, such as the string group \operatorname{String}(n), emerge in topological contexts as the 3-connected cover of \operatorname{Spin}(n), relevant to string theory and p_1-structures on bundles.^[103] These extend the double cover framework to kill the first two homotopy groups, providing models for topological string structures associated to orthogonal groups.^[103]

Principal homogeneous spaces

The Stiefel manifold V_{n,k}, consisting of all ordered k-tuples of orthonormal vectors in \mathbb{R}^n, serves as a key example of a principal homogeneous space for the orthogonal group O(n). It is diffeomorphic to the quotient space O(n)/O(n-k), where O(n-k) is the stabilizer subgroup of a standard frame under the transitive action of O(n). This realization endows V_{n,k} with the structure of the base space of a principal O(n-k)-bundle O(n) \to V_{n,k}, with the group acting freely on the total space via right multiplication.^[104] A fundamental fibration arises from projecting each orthonormal k-frame in V_{n,k} to its span, yielding the map V_{n,k} \to G_{n,k} to the Grassmannian of k-planes in \mathbb{R}^n, with fiber isomorphic to the orthogonal group O(k). This projection is a principal O(k)-bundle, where O(k) acts on the right by rotating frames within their span. For the special case k=1, V_{n,1} recovers the (n-1)-sphere S^{n-1}, illustrating how spheres embed as principal homogeneous spaces in this framework. In differential geometry, Stiefel manifolds relate closely to orthogonal frame bundles: the frame bundle of a Riemannian n-manifold is a principal O(n)-bundle over the manifold, with local sections corresponding to orthonormal frames, mirroring the global structure of V_{n,k} as a space of such frames in Euclidean space.^[105] Topologically, in the stable range where n is sufficiently large relative to k (specifically, for dimensions up to roughly n - k - 1), V_{n,k} is homotopy equivalent to O(n), reflecting the stabilization of homotopy groups under the inclusion of orthogonal groups.

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A continuous group (in fact, a Lie group) referred to as the orthogonal group in n dimensions on R and denoted as O(n, R ) or simply O(n).
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Orthogonal - Etymology, Origin & Meaning
Originating from Greek orthogonios meaning "right-angled," orthogonal (1570s, French) means pertaining to or depending upon the use of right angles.Missing: mathematics | Show results with:mathematics
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[PDF] Orthogonal matrices - How Euler Did It
On the subject of orthogonal matrices, he writes. The term ORTHOGONAL MATRIX was used in 1854 by. Charles Hermite (1822-1901) in the Cambridge and Dublin.
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https://math.ucsd.edu/~nwallach/newch1-g-w.pdf
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[PDF] Quadratic Forms Chapter I: Witt's Theories - UGA math department
Orthogonal Groups. 8.1. The orthogonal group of a quadratic space. Let V be a quadratic space. Then the orthogonal group O(V ) is, by definition, the group ...
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[PDF] 6 Orthogonal groups
We begin by considering some small cases. Let V be a vector space of rank n carrying a quadratic form Q of Witt index r, where δ n 2r is the dimension of.
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[PDF] Isometries of figures in Euclidean spaces - UCR Math Department
First of all, the set of all isometries of E is a group (sometimes called the Galileo group of E). It contains both the subgroups of orthogonal matrices and the ...
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[PDF] factoring euclidean isometries - UCSB Math
Abstract. Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection.
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[PDF] The Connectivity of SO(n) - Texas A&M University
Sep 18, 2013 · The special orthogonal group SO(n) is path-connected, meaning any matrix in SO(n) can be connected to the identity matrix by a path.
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[PDF] Rotations in 3-Dimensional Space† 1. Introduction
Orthogonal matrices for which det R = +1 are said to be proper rotations, while those with det R = −1 are said to be improper. Proper rotations have the ...
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[PDF] RES.18-011 (Fall 2021) Lecture 14: Symmetry Groups
Example 14.4. For a circle centered at the origin, every rotation or reflection will fix it, and so its symmetry group is all of O2. This group of symmetries is ...
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[PDF] Three-Dimensional Proper and Improper Rotation Matrices
A real orthogonal matrix with det R = −1 provides a matrix representation of an improper rotation. To perform an improper rotation requires mirrors. That is, ...
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[PDF] Rotation group - OU Math
Feb 19, 2010 · The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is ...
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[PDF] LIE GROUPS AND LIE ALGEBRAS - joseph redeker
Definition 2.7. The special orthogonal group is denoted SO(n) and defined as,. SO(n) := {A ∈ O(n) | det(A)=1}. Definition 2.8. The special unitary group is ...
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[PDF] Matrix Lie groups and their Lie algebras - Alen Alexanderian
Jul 12, 2013 · (d) The special orthogonal group SO(n): The proof that SO(n) is a matrix Lie group. combines the arguments for SL(n) and O(n) above. Remark 4.3 ...
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[PDF] GLn(R) AS A LIE GROUP Contents 1. Matrix Groups over R, C, and ...
Hence, the dimension of SO(n) is n + (n − 1) + ... +1= n(n−1). 2 . Page 10. 10. MAXWELL LEVINE. Tangent spaces can be explored in great detail, but first it ...
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[PDF] CS 4610/5335: Robotics - Representation of Orientation
Theorem: (Euler). Any orientation, R ∈ SO(3), is equivalent to a rotation about a fixed axis, k ∈ S2, through an angle, θ ∈ [−π, π]. Axis angle encodes an ...
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[PDF] CS@Cornell
Euler angles. • An object can be oriented arbitrarily. • Euler angles: stack up three coord axis rotations. • ZYX case: Rz(thetaz)*Ry(thetay)*Rx(thetax).
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[PDF] Topics in Representation Theory: Spin Groups
This is a reflection of the fact that Spin(n) is a double-covering of the group SO(n). Just as the adjoint action of the Lie algebra of Spin(n) on itself is ...
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Householder Matrix -- from Wolfram MathWorld
Hx=x-(a+a^(H))v. (3). Lehoucq (1996) independently gave an interpretation that still uses the formula Hx=x-2av , but ...
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What Is a Householder Matrix? - Nick Higham
Sep 15, 2020 · A Householder matrix is a rank- 1 perturbation of the identity matrix and so all but one of its eigenvalues are 1.
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[PDF] Special pure gradings on simple Lie algebras of types $E_6 ... - arXiv
Jul 4, 2025 · In odd dimension, the orthogonal group is isomorphic to a symplectic group, so the orthogonal group can be shown to be generated by reflections ...Missing: source:
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[PDF] A CONCISE PROOF OF SYLVESTER'S THEOREM
Thus, every symmetric quadratic form can be transformed into the canonical form, or in other words, every symmetric quadratic form is orthogonally congruent ...
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[PDF] Householder QR - CS@Cornell
Householder transformations are simple orthogonal transformations corre- sponding to reflection through a plane. Reflection across the plane orthogo-.
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[PDF] Classical homogeneous spaces 1. Rotations of spheres
Sep 25, 2010 · ... (unit) sphere Sn−1 in Rn. The preservation of the inner product does ... orthogonal group are the same. That is,. 1g ∈ GLn(R) : <gx, gy> ...
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[PDF] Lecture 5. Lie Groups
The special orthogonal group SO(n, k) ⊂ O(n, k) is the normal subgroup of orthogonal matrices of determinant 1. Proposition 5.1. (a) A ∈ O(n, k) if and only if.
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[PDF] 1 Representing Rotations
This is the strange topology of the rotation group SO (3). Remark 7. Euler angles notation: specifies a rotation by three angles (α,β,γ) as follows. First ...
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[PDF] Orthogonal groups in characteristic 2 acting on polytopes of high rank
Mar 13, 2018 · Corollary 1.5. For each integer k ⩾ 2, positive integer m, and ǫ ∈ {−, +}, the orthogonal group Oǫ(2m, F2k ) acts as the group of automorphisms ...
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[PDF] The geometry of orthogonal groups over finite fields
Let f,f0 be quadratic forms on vector spaces V,V 0 over K, respectively. An isometry σ : (V,f) −→ (V 0,f0) is an injective linear mapping from V to V 0 ...
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[PDF] Conjugacy Classes in Finite Orthogonal Groups
1 Orthogonal groups. In MAGMA the general orthogonal group in dimension n over the field k D GF.q/, where q is odd, consists of n n matrices A over k such ...Missing: Gram | Show results with:Gram
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[PDF] The Zassenhaus Decomposition for the Orthogonal Group
Abstract: The orthogonal group of a quadratic space over a field has been the focus of intensive research over the past fifty years.
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[PDF] Orthogonal Representations of Finite Groups
Oct 9, 2016 · In ordinary representation theory of finite groups one studies linear actions of finite groups on finite-dimensional vector spaces over a ...<|control11|><|separator|>
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[PDF] Volumes of Orthogonal Groups and Unitary Groups - arXiv
Nov 3, 2017 · This review article begins by presenting detailed key background knowledge about matrix integral. Then the volumes of orthogonal groups and ...
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[PDF] Orthogonal Stability - arXiv
Aug 26, 2022 · Abstract. A character (ordinary or modular) is called orthogonally stable if all non- degenerate quadratic forms fixed by representations ...
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[PDF] Reductive Groups
Abstract. These notes are a guide to algebraic groups, especially reductive groups, over a field. Proofs are usually omitted or only sketched.
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[PDF] (Linear) Algebraic Groups 1 Basic Definitions and Main Examples ...
We can create the orthogonal group OB = Oq = {A ∈ GL(V ) ... A connected linear algebraic group G contains a proper parabolic subgroup if and only if G is.
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[PDF] Math 249B. Root systems for split classical groups
Our work will also prove that the special orthogonal and symplectic groups really are semisimple (granting that they are smooth and connected, as was proved in ...
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[PDF] NOTES ON REPRESENTATIONS OF REAL REDUCTIVE GROUPS ...
We shall say that the real group G is split if a is a Cartan subalgebra of g. One can show that every connected complex reductive group has a unique split form ...
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[PDF] Reference sheet for classical roots systems - UC Berkeley math
Sep 21, 2023 · Mnemonic: “sln is the first semisimple Lie algebra you ever learn about, so its Dynkin diagram comes first in the alphabet.” • Weyl group: Sn.
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[PDF] Topics in Representation Theory: Maximal Tori and the Weyl Group
Theorem 1. If G is a compact connected Lie group and T is a maximal torus of G, then any element of G is conjugate to an element of T.
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[PDF] Weyl groups, affine
(ii) The hyperoctahedral group Bn, i.e., the group of ŉ ×n monomial matri- ces with non-zero entries in {1}. It contains S,, viewed as the group of ...
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Infinite dihedral group - Groupprops
Jan 12, 2024 · The infinite dihedral group is a particular group, unique up to isomorphism. This article provides specific information about it.
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[PDF] The Quaternions and the Spaces S3, SU(2), SO(3), and RP
The group SO(3) of rotations of R3 is intimately related to the 3- sphere S3 and to the real projective space RP3. The key to this relationship is the fact that ...Missing: double | Show results with:double
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[PDF] SU(2)'s Double-Covering of SO(3)
Dec 12, 2019 · We first point out the double covering, then we show the isomorphism of the Lie algebras of SU(2) and SO(3). We then explicitly find the ...Missing: topology RP^
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[PDF] Topology of SO(3) for Kids arXiv:2310.19665v1 [math.HO] 30 Oct 2023
Oct 30, 2023 · In the case of. SO(3) this is the group SU(2) of 2×2 unitary matrices with unit determinant. Topologically, this is the three-dimensional sphere ...
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[PDF] Exercise 1. Let PGL2(C) = GL2(C)/C∗I be the group of ...
σ : SU2/{±I} → SO3 = SO(V,−det). Prove that σ is an isomorphism (hint : show that diag(e−it/2,eit/2) ∈ SU2 acts ...<|control11|><|separator|>
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[PDF] From the Icosahedron to E8
Dec 22, 2017 · As we have seen, the rotational symmetry group of the icosahedron, A5 ⊂ SO(3), is double covered by the binary icosahedral group Γ ⊂ SU(2).<|separator|>
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What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)
Oct 9, 2015 · Over C the fundamental group of SO(n) is Z/2Z for n>2, Z when n=2, while the symplectic group is simply connected. Igor Rivin. – Igor Rivin.Topological structure of SO(n) as a product - MathOverflowWhy is the fundamental group of a compact Riemann surface not freeMore results from mathoverflow.net
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[PDF] Algebraic Topology - Cornell Mathematics
This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in ...
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[PDF] The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept ...
... SO(n) → SO(n+1) → SO(n+1)/SO(n) = Sn. By the long exact sequence in homotopy groups for this fibration we see that π1(SO(n)) → π1(SO(n + 1)) is an ...
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K -THEORY AND REALITY | The Quarterly Journal of Mathematics
M. F. ATIYAH; K -THEORY AND REALITY , The Quarterly Journal of Mathematics, Volume 17, Issue 1, 1 January 1966, Pages 367–386, https://doi.org/10.1093/qmat.
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[PDF] Looking for structure in the cobordism conjecture - arXiv
Jul 26, 2022 · The main example we consider is the White- head tower of the orthogonal group O (which coherently organises orientations, spin, string and ...
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[PDF] arXiv:1612.05664v1 [math.RA] 15 Dec 2016
Dec 15, 2016 · the p-th orthogonal group, and O(p, q) is the indefinite orthogonal group arising from a quadratic form with inertia (p, q). We also show ...
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Why is $O(n;k)$ not connected, and has four connected components?
Apr 16, 2018 · To see that there are exactly four connected components, you have to prove that O(n;k) is stable under the polar decomposition. Next prove that ...Indefinite orthogonal groups over p-adics - MathOverflowIntegral orthogonal group for indefinite ternary quadratic formMore results from mathoverflow.net
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[PDF] arXiv:1602.08760v2 [math.DG] 14 Jun 2017
Jun 14, 2017 · Let SO+(p, q) denote the identity connected component of the real orthogonal group with signature (p, q). We give a complete description of the.
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[PDF] arXiv:2308.13947v2 [math.AT] 29 Aug 2023
Aug 29, 2023 · SO(p) × SO(q) is a maximal compact subgroup of SO+(p, q). Proof. Again we begin by proving O(p) × O(q) is a maximal compact subgroup of O(p, q) ...
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[PDF] 1 The Classical Groups - UCSD Math
The special orthogonal group over F is the subgroup SO(n,F) = O(n,F) ∩ SL(n,F) of O(n,F). The indefinite special orthogonal groups are the groups SO(p,q) = O(p ...Missing: mathematics | Show results with:mathematics
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Iwasawa decomposition of the pseudo-orthogonal group
Apr 19, 2014 · Indeed, it is easy to write out the Iwasawa decomposition for orthogonal groups SO(p,q), say with p≥q. By a change of variables, ...Compatible Iwasawa decomposition for embedding of the ...Orbits of the maximal compact subgroup on the light cone for $p ...More results from mathoverflow.net
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[PDF] arXiv:1212.0607v1 [math.RT] 4 Dec 2012
Dec 4, 2012 · These elements are constructed in accordance with the Iwasawa decomposition of the real rank one indefinite orthogonal Lie algebra. We also.
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Lorentz Group -- from Wolfram MathWorld
The Lorentz group is the group L of time-preserving linear isometries of Minkowski space R^((3,1)) with the Minkowski metric dtau^2=-(dx^0)^2+(dx^1)^2+(dx^2)^2 ...<|control11|><|separator|>
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[PDF] 15.3 More about Orthogonal groups 15.4 Spin groups in small ...
(1) It is the set of positive definite 2 dimensional subspaces of R2,n. (2) It is the norm 0 vectors ω of PC2,n with (ω, ¯ω) = 0. (3) It is the vectors x + iy ∈ ...Missing: fundamental | Show results with:fundamental
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A new formula for the spinor norm - AIP Publishing
Jun 1, 1978 · A formula is established, valid for all real pseudo‐orthogonal groups, permitting an easy computation of the spinor norm of any ...
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[PDF] Quadratic spaces and Orthogonal groups
Definition 1.4. We respectively define the orthogonal group and special orthogonal group of. a quadratic space (V,Q) as the groups. OV = {𝜎 ∈ Aut(V ) | 𝜎 is an ...
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[PDF] Quadratic Forms, Clifford Algebras and Spinors
Special Orthogonal Groups and Spin. Groups. Let (V; q) be a quadratic form and let O(q) be its orthogonal group. We denote by Aut g. K(C) the group of ...
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https://www.csun.edu/~sungjin/HasseMinkowski.pdf
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[PDF] arXiv:1810.04228v1 [math.AG] 9 Oct 2018
Oct 9, 2018 · The identification of quadratic forms with torsors for the orthogonal groups is classical, cf. [RV06, Section 5] or [Wen17, Section 2] ...
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[PDF] A QUICK NOTE ON ORTHOGONAL LIE ALGEBRAS
Definition 1. The special orthogonal Lie algebra of dimension n ≥ 1 over R is defined as so(n,R) . = {A ∈ gl(n,R) | A. > + A = 0}. It is a vector subspace of ...
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[PDF] DIFFERENTIAL GEOMETRY HW 12 3 Find the Lie algebra so(n) of ...
dimension of SO(n) and, therefore, so(n), so(n) consists precisely of the skew-symmetric n × n matrices. D. 2. Show that x : R2 → R4 given by x(θ, φ) = 1.
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[PDF] Classification of Semisimple Lie Algebras
Weyl group of any root system is generated by the set of reflections by elements of its base. Furthermore, one can show that every element of R is part of ...<|separator|>
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[PDF] From Lie Algebras to Chevalley Groups - White Rose eTheses Online
So, given a torus, we can find a basis for L on which the entire torus acts diagonally. This enables us to decompose the Lie algebra into its maximal torus and ...
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[PDF] Math 508 – Lie Algebras (lecture notes)
Apr 21, 2023 · 2.4 Exercise. Consider two cases for the orthogonal Lie algebra son. 1. When n = 2l + 1 is odd, let. S =.<|control11|><|separator|>
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[PDF] The Killing Form, Reflections and Classification of Root Systems 1 ...
The Killing form, defined as K(X, Y ) = Tr(ad(X) ◦ ad(Y )) on g, is used to define an inner product on t∗, and is strictly negative definite on g for compact ...
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[PDF] 10 Killing form and Cartan's criterion
One answer is Cartan's criterion, which implies that the Killing form on a simple complex Lie algebra (a symmetric invariant bilinear form) is non-degenerate.<|control11|><|separator|>
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On Subgroups of the Orthogonal Group - jstor
Samelson [5; 6], in this regard, have proved that if dim 0(n) > dim G ? dim 0(n - 1) then G is conjugate to the standard subgroup SO(n- 1) except n =4, 8 and.Missing: sources | Show results with:sources
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[PDF] classifying the finite subgroups of so3 - UChicago Math
Aug 29, 2020 · We prove that all finite subgroups of SO3 are isomorphic to either a cyclic group, a dihedral group, the tetrahedral group, the octahedral group ...
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[PDF] arXiv:2112.08079v3 [math.DG] 21 Jul 2022
Jul 21, 2022 · They are parametrized by the structure group of conformal geometry, CO(n) ⋉ Rn, where CO(n) = O(n) × R+ is the conformal orthogonal group.
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[PDF] The orthosymplectic supergroup in harmonic analysis - arXiv
Feb 3, 2012 · In classical harmonic analysis, the orthogonal group O(m) corresponds to the group of isometries of Rm which stabilize the origin. We use ...
[88]
Definition of orthosymplectic supergroups - MathOverflow
Mar 11, 2017 · The Lie supergroup OSP(m|n) is defined by OSP(m|n)={M∈GL(m|n):m=2p,MstHM=H},. where H=(Q00In),Q=(0Ip−Ip0),. In is the identity matrix of order n ...
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[PDF] The Pin and Spin Groups - University of Pennsylvania
Nov 9, 2013 · In fact, this is also true for the orthogonal group, ... Proposition 1.21 For all p, q ≥ 0, the groups Pin(p, q) and Spin(p, q) are double covers ...
[90]
General linear group inclusion - Mathematics Stack Exchange
Sep 11, 2023 · I think you get such an embedding by letting g∈GL(n,F) act as g on a totally singular n-dimensional subspace of the 2n-dimensional orthogonal ...
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[PDF] finite subgroups of the orthogonal group in three dimensions and ...
The subgroups of Dn depend on whether n is odd or even and are either cyclic or dihedral. Case 1: When n is odd. In this case, the subgroups of Dn of order 2 ...<|separator|>
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[1108.2602] On finite groups acting on spheres and finite subgroups ...
Aug 12, 2011 · We discuss finite groups acting on low-dimensional spheres, comparing with the finite subgroups of the corresponding orthogonal groups, and also ...
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https://www.worldscientific.com/doi/10.1142/9789811286605_0002
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crystallographic groups and their mathematics - Project Euclid
Jun 14, 1979 · We return to our finite subgroups of the general linear group. For ob- vious reasons, it is desirable to move such a group into the orthogonal.<|separator|>
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[PDF] Algebraic Groups, Lie Groups, and their Arithmetic Subgroups
Apr 1, 2011 · This work is a modern exposition of the theory of algebraic groups, Lie groups, and their arithmetic subgroups.
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[PDF] Essential dimension of the spin groups in characteristic 2
The spin group Spin(n) is the simply connected double cover of SO(n). It was a surprise when Brosnan, Reichstein, and Vistoli showed that the essential ...
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[PDF] Meet spin geometry Spin structures and spin manifolds
Spin geometry involves spin structures on vector bundles, related to the Lie group Spin(n), which is a 2-fold covering of SO(n). Two spin structures can be ...
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[PDF] Double covers of pseudo-orthogonal groups - prof. Andrzej Trautman
For every n ∈ N, one has the orthogonal group On, its connected component SOn, the spin group Spinn, two pin groups Pin+ n and Pin. - n : if u ∈ Rn ∩ Pint.
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[PDF] Chapter 20 Manifolds Arising from Group Actions - UPenn CIS
n. (1) is a symmetric space of noncompact type. The symmetric space H. + n. (1) = SO0(n,1)/SO(n) turns out to be dual, as a symmetric space, to. S n. = SO(n + 1)/ ...
[100]
COHOMOGENEITY ONE SPECIAL LAGRANGIAN SUBMANIFOLDS ...
We describe the cotangent bundle of the n-sphere Sn ∼= SO(n + 1)/SO(n) by. T ... submanifolds L invariant under SO(n) are diffeomorphic to I × Sn−1.
[101]
[PDF] What is the order of the Weyl group of E8? We'll do this by 4 different
For E8, the fundamental invariants are of degrees 2,8,12,14,18,20,24,30 (primes +1). (3) This one is actually an honest method (without quoting weird facts).Missing: quotients | Show results with:quotients
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The birth of E8 out of the spinors of the icosahedron - Journals
Jan 1, 2016 · They form a subgroup of the Pin group and constitute a double cover of the special orthogonal group, called the Spin group. Clifford algebra ...<|control11|><|separator|>
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[PDF] String structures associated to indefinite Lie groups - arXiv
Mar 31, 2019 · We extend the description of String structures from connected covers of the definite-signature orthogonal group O(n) to the indefinite- ...
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[PDF] CHARACTERISTIC CLASSES
Stiefel manifold bundle is nothing but a k-tuple of linearly independent cross-sections of the vector bundle f. Now suppose that the base space B is a CW ...
[105]
[PDF] Chapter 3 - Vector bundles and principal bundles
The orthogonal group O(n) acts freely and fiberwise transitively on this space, endowing Fr(ξ) with the structure of a principal O(n)-bundle. Providing a vector ...