Vipul: Created page with "==Definition== Suppose $n$ is a natural number. The special orthogonal group of degree $n$ over the reals, denoted $SO(n,\R)$ , is a [[Lie group]..."

2011-07-29T01:36:55Z

Created page with "==Definition== Suppose <math>n</math> is a natural number. The special orthogonal group of degree <math>n</math> over the reals, denoted <math>SO(n,\R)</math>, is a [[Lie group]..."

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==Definition==

Suppose <math>n</math> is a natural number. The special orthogonal group of degree <math>n</math> over the reals, denoted <math>SO(n,\R)</math>, is a [[Lie group]] that can be defined concretely as the group of <math>n \times n</math> matrices with real entries whose determinant is 1 and whose product with the transpose is the identity matrix:

<math>SO(n,\R) = \{ A \in \operatorname{Mat}_n(\R) : \operatorname{det} A = 1, \ \operatorname{and} \ A^T = A^{-1} \}</math>

==Particular cases==

{| class="sortable" border="1"
! Value of <math>n</math> !! Dimension of <math>SO(n,\R)</math> (in general equals <math>n(n+1)/2 - 1</math>) !! Description of <math>SO(n,\R)</math>, or rather, its underlying topological space
|-
| 1 || 0 || [[one-point space]]
|-
| 2 || 1 || [[circle]] <math>S^1</math>
|-
| 3 || 3 || [[real projective three-dimensional space]] <math>\mathbb{P}^3(\R)</math>
|-
| 4 || 6 || [[special orthogonal group:SO(4,R)]]
|-
| 5 || 10 || [[special orthogonal group:SO(5,R)]]
|}

==Algebraic topology==

===Homology===

{{further|[[homology of special orthogonal group over reals]]}}

===Cohomology===

{{further|[[cohomology of special orthogonal group over reals]]}}

===Homotopy===

{{further|[[homotopy of special orthogonal group over reals]]}}

Special orthogonal group over reals - Revision history

Vipul: Created page with "==Definition== Suppose n is a natural number. The special orthogonal group of degree n over the reals, denoted SO(n,\R), is a [[Lie group]..."

Vipul: Created page with "==Definition== Suppose $n$ is a natural number. The special orthogonal group of degree $n$ over the reals, denoted $SO(n,\R)$ , is a [[Lie group]..."