General affine group: Difference between revisions

Revision as of 18:54, 31 May 2012

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 or affine general linear group of degree $n$ over $K$ , denoted $G A (n, K)$ , $G A_{n} (K)$ , $A G L (n, K)$ , or $A G L_{n} (K)$ , is defined as the external semidirect product of the vector space $K^{n}$ by the general linear 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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 ===In terms of dimension===
-Let <math>n</math> be a [[natural number]] and <math>k</math> be a [[field]]. The '''general affine group''' or '''affine general linear group''' of degree <math>n</math> over <math>k</math>, denoted <math>GA(n,k)</math>, <math>GA_n(k)</math>, <math>AGL(n,k)</math>, or <math>AGL_n(k)</math>, is defined as the [[external semidirect product]] of the vector space <math>k^n</math> by the [[defining ingredient::general linear group]] <math>GL(n,k)</math>, acting by linear transformations.
+Let <math>n</math> be a [[natural number]] and <math>K</math> be a [[field]]. The '''general affine group''' or '''affine general linear group''' of degree <math>n</math> over <math>K</math>, denoted <math>GA(n,K)</math>, <math>GA_n(K)</math>, <math>AGL(n,K)</math>, or <math>AGL_n(K)</math>, is defined as the [[external semidirect product]] of the vector space <math>K^n</math> by the [[defining ingredient::general linear 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.
+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>.
+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>.