I stumbled over a statement on Wikipedia <http://en.wikipedia.org/wiki/Duality_%28projective_geometry%29> and would like to ask how this could possibly be true. It states the following



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> The projective planes $PG(2,K)$ for any division ring $K$ are self-dual.
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Is this really true? Or rather: What is wrong with the following argument that (if correct) would contradict the statement:


For a finite dimensional left vector space $V$ over a division ring $K$, we have the dual left vector space $V^{\star}$ over $K^{op}$ (the opposite division ring of $K$). Now, it is certainly true that $P(V^{\star})$ is isomorphic to the dual of $P(V)$. Thus, $P(V)$ is isomorphic to its dual if and only if $P(V) \cong P(V^{\star})$. Since $V$ and $V^{\star}$ have the same dimension, does that not necessarily imply that $K$ and $K^{op}$ are isomorphic by the Second Fundamental Theorem of projective Geometry? 


In fact, isn't $PG(2,K)$ (or $PG(n,K)$ for any $n \geq 2$) being self-dual simply eqvuialent to $K \cong K^{op}$ (which is of course is not always the case). I also recall that I have read this somewhere.


So, to put it in one line: What is wrong? The statement on Wikipedia or my reasoning?




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EDIT: I found some course notes in which it is also claimed that $PG(n,K)$ is self-dual iff $K$ is self-dual (see 6.1 in <http://www.maths.qmul.ac.uk/~pjc/pps/pps6.pdf>). This does contradict what Wikipedia says, doesn't it? 



Is Wikipedia correct about desarguesian projective planes being self-dual?
I think you are right. There is a theorem that asserts that if two projective spaces $P(V,F)$ and $P(V',F')$ are isomorphic then the skew fields $F$ and $F'$ are isomorphic. Thus if $P(V,F)$ and $P(V^\*,F^{op})$ are isomorphic then $F\cong F^{op}$ as skew fields. 


I recommand these course notes "Essential Concepts of Projective Geometry" available here (the author gives a lot of references):
<http://math.ucr.edu/~res/progeom/>
in particular look at appendix C. 


