polarities and forms
To see this, let be a duality. We may express this as an order preserving map . Then by the fundamental theorem of projective geometry it follows is induced by a bijective semi-linear transformation .
Now, if is a non-degenerate sesquilinear form. Then define
which is semi-linear, as is sesquilinear, and bijective, since is non-degenerate. Therefore induces an order preserving bijection , that is, a duality. ∎
We write for the image of the induced duality of a non-degenerate sesquilinear form . Notice that . (Although the form may not be reflexive, we still use the notation, but we now demonstrate that we can indeed specialize to the reflexive case.) Notice then that
Every polarity gives rise to a reflexive non-degenerate sesquilinear form, and visa-versa.
Let be the sesquilinear form induced by the polarity . Then suppose we have such that . So . But has order 2 so . But this implies so is reflexive.
From the fundamental theorem of projective geometry it follows if then every order preserving map is induced by a semi-linear transformation of . In similar fashion we have
, meaning that every order reversing map can be decomposed as a where is induced from a semi-linear transformation and is a polarity.
Let be any duality of . Then is order preserving. Thus is a projectivity so by the fundamental theorem of projective geometry is induced by a semi-linear transformation . Therefore has index 2 in . Finally it suffices to provide any polarity of to prove . For this use any reflexive non-degenerate sesquilinear form. ∎
|Title||polarities and forms|
|Date of creation||2013-03-22 15:58:13|
|Last modified on||2013-03-22 15:58:13|
|Last modified by||Algeboy (12884)|