Transvection

2020 Mathematics Subject Classification: Primary: 20-XX [MSN][ZBL]

A transvection is a linear mapping $f$ of a (right) vector space $V$ over a skew-field $K$ with the properties

$$\def\rk{\textrm{rk}\;}\rk(f-E) = 1\quad\textrm{and}\quad \textrm{Im}(f - E)\subseteq \ker(f-E),$$ where $E$ is the identity linear transformation. A transvection can be represented in the form

$$\def\a{\alpha} f(x) = x+a\a(x),$$ where $a\in V$, $\a\in V^*$ and $\a(a) = 0$.

The transvections of a vector space $V$ generate the special linear, or unimodular, group $\def\SL{\textrm{SL}}\SL(V)$. It coincides with the commutator subgroup of $\def\GL{\textrm{GL}}\GL(V)$, with the exception of the cases when $\dim V = 1$ or $\dim V = 2$ and $K$ is the field of two elements. If $K$ is a field, then $\SL(V)$ is the group of matrices with determinant 1. In the general case (provided that $\dim V \ne 1$), $\SL(V)$ is the kernel of the epimorphism

$$\GL(V) \to K^*/[K^*,K^*],$$ which is called the Dieudonné determinant (cf. Determinant).

In the projective space $\def\P{\mathbb{P}}\P(V)$, whose points are the $1$-dimensional subspaces of $V$, a transvection $f$ as above induces a (projective) transvection with $aK$ as centre and $\ker(f-E)$ as axis. If one takes $\ker(f-E)$ to be a hyperplane at infinity in $\P(V)$, such a transvection induces a translation $x\mapsto x+b$ in the remaining affine space (interpreted as a linear space). See also Shear.

References

[Di]	J.A. Dieudonné, "La géométrie des groupes classiques", Springer (1955) MR0072144 Zbl 0067.26104