Algebraic projective geometry by the late J. G. Semple, G. T. Kneebone

By the late J. G. Semple, G. T. Kneebone

First released in 1952, this e-book has confirmed a helpful advent for generations of scholars. It presents a transparent and systematic improvement of projective geometry, development on ideas from linear algebra.

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1 Some representation theory Recall the definition of a group representation: Definition (Representation). A (complex) representation (π, V ) of a group G on a complex vector space V (with a chosen basis identifying V Cn ) is a homomorphism π : G → GL(n, C) This is just a set of n by n matrices, one for each group element, satisfying the multiplication rules of the group elements. n is called the dimension of the representation. We are mainly interested in the case of G a Lie group, where G is a differentiable manifold of some dimension.

Comdf we find puting the derivative f (θ) = dθ f (θ + ∆θ) − f (θ) ∆θ (f (∆θ) − 1) = f (θ) lim (using the homomorphism property) ∆θ→0 ∆θ = f (θ)f (0) f (θ) = lim ∆θ→0 Denoting the constant f (0) by c, the only solutions to this differential equation satisfying f (0) = 1 are f (θ) = ecθ Requiring periodicity we find f (2π) = ec2π = f (0) = 1 which implies c = ik for k ∈ Z, and f = πk for some integer k. 18 The representations we have found are all unitary, with πk taking values in U (1) ⊂ C∗ . The complex numbers eikθ satisfy the condition to be a unitary 1 by 1 matrix, since (eikθ )−1 = e−ikθ = eikθ These representations are restrictions to the unit circle U (1) of irreducible representations of the group C∗ , which are given by πk : z ∈ C∗ → πk (z) = z k ∈ C∗ Such representations are not unitary, but they have an extremely simple form, so it sometimes is convenient to work with them, later restricting to the unit circle, where the representation is unitary.

In terms of adjoints, this condition becomes Lv, Lw = v, L† Lw = v, w so L† L = 1 or equivalently L† = L−1 In matrix notation this first condition becomes n n (L† )jk Lkl = k=1 Lkj Lkl = δjl k=1 which says that the column vectors of the matrix for L are orthonormal vectors. Using instead the equivalent condition LL† = 1 we find that the row vectors of the matrix for L are also orthonormal. Since such linear transformations preserving the inner product can be composed and are invertible, they form a group, and some of the basic examples of Lie groups are given by these groups for the cases of real and complex vector spaces.

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