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From: Mike on 14 Jul 2010 03:04 According to the wikipedia, orthogonal group over any field is an algebraic group, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix. Algebraic group is defined to be a group which is an algebraic variety, I don't really know much about algebraic variety but I guess it is something similar to closed algebraic subset of k^n where k is an algebraically closed field. I suppose that real orthogoanl group O(n,R) is not an algebraic subset of C^n where C is a complex field. I guess O(n,C) is a algebraic group, but I don't know why O(n,R) is an algebraic group. Could anyone explain about this?
From: Hagen on 14 Jul 2010 00:19
> According to the wikipedia, orthogonal group over any > field is an > algebraic group, because the condition that a matrix > be orthogonal, > i.e. have its own transpose as inverse, can be > expressed as a set of > polynomial equations in the entries of the matrix. > > Algebraic group is defined to be a group which is an > algebraic > variety, I don't really know much about algebraic > variety but I guess > it is something similar to closed algebraic subset of > k^n where k is > an algebraically closed field. > > I suppose that real orthogoanl group O(n,R) is not an > algebraic subset > of C^n where C is a complex field. I guess O(n,C) is > a algebraic > group, but I don't know why O(n,R) is an algebraic > group. > > Could anyone explain about this? > > The definition of an algebraic group you took from Wikipedia is not correct: an algebraic group over the field k (not necessarily algebraically closed) is: + an algebraic variety X over k, + a group ... + ... such that the group operations X x X --> X, (x,y) --> xoy X --> X, x --> x^(-1) are morphisms of algebraic varieties over k. Note that the product X x X is an algebraic variety over k. The orthogonal group O(n,k) over k is an algebraic subset of k^{n^2} because it consists of those matrices, that satisfy A^t * A = A * A^t = E, which is a set of finitely many polynomial equations. Consequently and in modern language O(n,K) = Spec ( k[x1,x2,...,xn^2] / I I := ideal of polynomials appearing in A^t * A = A * A^t = E. Thus O(n,k) even is an affine algebraic set. To show that O(n,k) is a variety one has to prove that the ideal I is a prime ideal. The group operation (matrix multiplication) obviously is a morphism of k-varieties, because it is given by polynomials with coefficients in k in the matrix coefficients as variables. Matrix inversion is a morphism of k-varieties because the coefficients of the inverse are (linear) polynomials in the coefficients of the matrix itself. H |