Projective space

From Example Problems
Jump to navigation Jump to search

In mathematics, projective spaces are a fundamental construction, obtained from a vector space over an arbitrary division ring, in particular over a field. They generalise the notion of projective plane, which is constructed from a three-dimensional vector space. Projective spaces are essential to algebraic geometry through the rich field of projective geometry developed in the nineteenth century, but also in the constructions of the modern theory (based on graded algebras). Projective spaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory.

The basic construction, given a vector space V over a division ring K, is to form the set of equivalence classes of non-zero vectors in V under the relation of scalar proportionality: we consider v to be proportional to w if v = cw with c in K non-zero. This idea goes back to mathematical descriptions of perspective. If K is the real numbers, and V has dimension n, then the projective space P(V) - which we can talk about as the space of lines through the zero element 0 of V - carries a natural structure of a compact smooth manifold of dimension n − 1. It is also highly symmetric, since any linear automorphism of V gives rise to a symmetry of P(V). These in the classical examples identify with 'perspectivity' and 'projectivity' transformations described geometrically, and account for the name. The group of these symmetries is the quotient of the general linear group of V by the subgroup of non-zero scalar multiples of the identity.

The use of projective spaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the projective space associated to a four-dimensional real vector space. In that way geometrical ideas introduced by Poncelet and others become part of a theory founded on linear algebra. The part of a projective space not 'at infinity' is called affine space; but the symmetries of P(V) do not respect that division. Use of a basis of V allows, if required, the introduction of homogeneous co-ordinates for the handling of concrete calculations.

Use of vector spaces over the field of complex numbers gives rise to different manifolds, also used by geometers. There are good reasons for using them, in order to get a theory about intersections of algebraic varieties with predictable properties. In the theory of Alexander Grothendieck there are reasons for applying the construction outlined above rather to the dual space V*.

Morphisms

Projective linear maps between two projective spaces over the same field, say, P(V) and P(W), have the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T*:[\mathbf{v}]\mapsto [T(\mathbf{v})],}

where T is an element of L(V,W), the space of linear maps between V and W, v is an element of V, and we consider the equivalence classes under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. Two linear maps S and T in L(V,W) induce the same map between P(V) and P(W) iff they differ by a scalar multiple of the identity, that is if T=kS for some k ≠ 0. Thus if one identifies the scalar multiples of the identity map with the underlying field, the set of morphisms from P(V) to P(W) is simply P(L(V,W)).

The automorphisms, the invertible projective linear maps from a projective space to itself, can be described more concretely. Consider the invertible linear maps from the underlying vector space to itself; these form a group, and the projective linear maps are an image of the group, under the map Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T \mapsto T* } . Aut(P(V)) is the quotient group Aut(V)/Z(V), where Z(V) is again the group of nonzero scalar multiples of the identity, which is the kernel of the mapping. Z(V) is the center of Aut(V). This is why such quotient groups as known in general as projective linear groups.

See also

fr:espace projectif