Orthogonal group | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-05

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Orthogonal_group


00:03:17 1 Name
00:04:06 2 In even and odd dimension
00:04:40 3 Over the real number field
00:05:37 3.1 Presentation as real algebraic variety
00:06:14 3.2 Geometric interpretation
00:10:22 3.3 Maximal tori and Weyl groups
00:10:40 3.4 Low-dimensional topology
00:10:44 3.5 Fundamental group
00:11:10 3.6 Homotopy groups
00:11:44 3.6.1 Relation to KO-theory
00:14:12 3.6.2 Computation and interpretation of homotopy groups
00:16:36 3.6.2.1 Low-dimensional groups
00:17:06 3.6.2.2 Lie groups
00:17:11 3.6.2.3 Vector bundles
00:17:20 3.6.2.4 Loop spaces
00:17:51 3.6.3 Interpretation of homotopy groups
00:21:27 3.6.4 Whitehead tower
00:21:47 4 Over the complex number field
00:23:30 5 Over finite fields
00:23:41 6 The Dickson invariant
00:23:58 7 Orthogonal groups of characteristic 2
00:24:18 8 The spinor norm
00:24:32 9 Galois cohomology and orthogonal groups
00:24:46 10 Lie algebra
00:25:03 11 Related groups
00:25:19 11.1 Lie subgroups
00:25:43 11.2 Lie supergroups
00:25:47 11.2.1 Conformal group
00:25:56 11.3 Discrete subgroups
00:26:14 11.4 Covering and quotient groups
00:26:21 12 Principal homogeneous space: Stiefel manifold
00:26:49 13 See also
00:27:06 13.1 Specific transforms
00:27:41 13.2 Specific groups
00:28:05 13.3 Related groups
00:29:18 13.4 Lists of groups
00:30:06 14 Notes
00:34:04 15 Citations
00:35:32 16 References
00:37:14 17 External links
00:37:51 Galois cohomology and orthogonal groups
00:39:59 Lie algebra
00:40:55 2k + 1, while in even dimension Dr, where n
00:42:47 Related groups
00:43:11 USp(2n) and USp(n) ⊂ U(n) ⊂ O(2n) are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, U(n)/O(n) is the Lagrangian Grassmannian.
00:43:31 Lie subgroups
00:45:02 Lie supergroups
00:46:27 Conformal group
00:47:13 O(2k + 1) × R∗, where R∗
00:47:48 CO(n) ∩ GL+(n)
00:48:02 Discrete subgroups
00:49:01 Covering and quotient groups
00:50:25 Principal homogeneous space: Stiefel manifold
00:51:08 See also
00:51:18 Specific transforms
00:51:33 Specific groups
00:51:47 Related groups
00:52:02 Lists of groups
00:52:16 Notes



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SUMMARY
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In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose.
An important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1. This group is also called the rotation group, because, in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2), SO(3) and SO(4).
The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field.  In particular, when the bilinear form is the scalar product on the vector space F n of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, denoted O(n, F ), is the set of n×n orthogonal matrices with entries from F, with the group operation of matrix multiplication. This is a subgroup of the general linear group GL(n, F ) given by

  
    
      
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