General affine group: Difference between revisions

Revision as of 15:19, 16 March 2009

Template:Field-parametrized linear algebraic group

Definition

In terms of dimension

Let $n$ be a natural number and $k$ be a field. The general affine group of order $n$ over $k$ , denoted $G A (n, k)$ or $G A_{n} (k)$ , is defined as the external semidirect product of the vector space $k^{n}$ by the group $G L (n, k)$ , acting by linear transformations.

While $G A (n, k)$ cannot be realized as a subgroup of $G L (n, k)$ , it can be realized as a subgroup of $G L (n + 1, k)$ in a fairly typical way: the vector from $k^{n}$ is the first $n$ entries of the right column, the matrix from $G L (n, k)$ is the top left $n \times n$ block, there is a $1$ in the bottom right corner, and zeroes elsewhere on the bottom row.

In terms of vector spaces

Let $V$ be a $k$ -vector space (which may be finite- or infinite-dimensional). The general affine group of $V$ , denoted $G A (V)$ , is defined as the external semidirect product of $V$ by $G L (V)$ .

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-{{field-parametrized group property}}
+{{field-parametrized linear algebraic group}}
-{{natural number-parametrized group property}}
 ==Definition==
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 Let <math>n</math> be a [[natural number]] and <math>k</math> be a [[field]]. The '''general affine group''' of order <math>n</math> over <math>k</math>, denoted <math>GA(n,k)</math> or <math>GA_n(k)</math>, is defined as the [[external semidirect product]] of the vector space <math>k^n</math> by the group <math>GL(n,k)</math>, acting by linear transformations.
+While <math>GA(n,k)</math> cannot be realized as a subgroup of <math>GL(n,k)</math>, it ''can'' be realized as a subgroup of <math>GL(n+1,k)</math> in a fairly typical way: the vector from <math>k^n</math> is the first <math>n</math> entries of the right column, the matrix from <math>GL(n,k)</math> is the top left <math>n \times n</math> block, there is a <math>1</math> in the bottom right corner, and zeroes elsewhere on the bottom row.
 ===In terms of vector spaces===
 Let <math>V</math> be a <math>k</math>-vector space (which may be finite- or infinite-dimensional). The general affine group of <math>V</math>, denoted <math>GA(V)</math>, is defined as the external semidirect product of <math>V</math> by <math>GL(V)</math>.