# Unitary geometry

• Martin Götzky
Chapter
Part of the NATO ASI Series book series (ASIC, volume 333)

## Abstract

Let V=Vn(K,fα) be an n-dimensional [regular] unitary vector space over a skewfield K of characteristic distinct from 2, where fα is an α-Hermitian form defined on V and n ⊥ 3. Let U = Un(K,fα) denote the associated unitary group. For each transformation πεU define the path B(π): = = V (π-Id) and the fix F(π): = kernel (π-Id). An element π ε U is called simple if dim B(π) equals 1. Let E be a point (i.e. a 1-dimensional subspace of V) of the projective space PV over V. Then the set U(E) of simple transformations with path E together with the identity is a subgroup of U representing in U the point E of the projective space. Then PV is equipped with a polarity and an orthogonality relation ⊥ induced by fα. The following properties hold:
1. (*)

Given points E1,…,Em of PV and σ i ε U(Ei)\{Id}. Then E1, …, Em are linearly dependent if and only if σ 1… σm is a product of fewer than m simple transformations.

2. (**)

Given σi ε U(Ei)\{Id} (i=1,2) where E1 and E2 are distinct. Then σ1 commutes with σ2 if and only if E1 ist orthogonal to E2.

Now consider the pair (U,S) where S is the set of simple transformations of the unitary group U. Using the properties (*) and (**) one can reconstruct the projective space PV and the polarity ⊥ In order to reconstruct the vector space V one needs Desargues’ axiom (D). This causes problems if n = 3.

Section 2 outlines a group-theoretical characterization of “generated unitary groupes” (U,S) for n > 3. Section 3 states an alternative characterization. Finally, in Section 4 some remarks on the effect of the choice of generators on the structure of the unitary group are made.

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