Matrix Groups over

Theorem. Let S be a connected solvable matrix group over an algebraically closed field. Then there is a basis in which all elements ofS are upper triangular (i.e., zero below the diagonal). 

Theorem. Let S bed connected solvable matrix group over any field. Then the unipotent elements in S form a normal subgroup which contains all commutators. 

Theorem. Let Nbea connected nilpotent algebraic matrix group over a perfect field. Then the separable and unipotent elements form closed subgroups N, and Nu of which N is the direct product. 

Let S be a connected solvable algebraic matrix group over a perfect field. If the separable elements form a subgroup, show that S is nilpotent. [S, is normal and S, n Su is trivial, so Ss and Su commute and S = S,x Su. Then S, is connected and hence abelian.]... 

Let G be a connected algebraic matrix group over an algebraically closed field k with char(Jtc) = 0. Generalizing a well-known result for finite groups, one naturally asks which G are such that all representations are sums of irreducible representations. This has a quite simple answer in terms of the structure of G. 

Borel, A. Linear Algebraic Groups (New York Benjamin, 1969). Mainly structure theory for algebraic matrix groups over algebraically closed fields, with some discussion of other fields. 

The columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

We call G° the connected component of G. Unlike algebraic matrix groups, the G here need not have the other f( A isomorphic to A0 this fails in our introductory example of p3 over the reals. 

Proof. The closure of N over k is still nilpotent, and by (9.2) the decomposition of elements takes place in k, so we may assume k is algebraically closed. The center of N is an abelian algebraic matrix group to which (9.3) applies. If the set Ns is contained in the center, it will then be a closed subgroup, and the rest is obvious from the last theorem. Thus we just need to show Nt is central. 

The smoothness of algebraic matrix groups is a property not shared by all closed sets in /c". To see what it means, take fc = fc and let 5 fc" be an arbitrary irreducible closed set. Let s be a point in S corresponding to the maximal ideal J in k[S]. If S is smooth, n si k = O si /J us) has fc-dimension equal to the dimension of S. (This would in general be called smoothness at s.) If S is defined by equations fj = 0, the generators and relations for OUS] show that S is smooth at s iff the matrix of partial derivatives (dfj/dXi)(s) has rank n — dim V. Over the real or complex field this is the standard Jacobian criterion for the solutions of the system (f = 0) to form a C or analytic submanifold near s. For S to be smooth means then that it has no cusps or self-crossings or other singularities . 

To understand this, take the matrix group G — GL2, with H the upper triangular group. Here G acts on k1 = kei ke2, and H is the stabilizer of ev In fact G acts transitively on the set of one-dimensional subspaces and since H is the stabilizer of one of them, the coset space is the collection of those subspaces. But they form the projective line over k, which is basically different from the kind of subsets of fc" that we have considered. In the complex case, for instance, it is the Riemann sphere, and all analytic functions on it are constant whereas on subsets of n-space we always have the coordinate projection functions. 

In this chapter and the last, we have established the reasonable properties one would expect quotients to have. In particular, the abelian affine group schemes over a field form an abelian category (Ex. 12). For the reasons indicated in (15.3), this is not true for algebraic matrix groups. 

Since Gm is central in GL , a further step can be taken here, constructing a map Hl(k,/k, PGLJ - H2(kjk, Gm) this map is actually injective (/ (GLJ is trivial). These injections exist for each n one can show that their images exhaust H2 kt/k, Gm), and that classes for different n have the same image iff they yield the same element in the Brauer group (i.e., are matrix algebras over the same division ring). 

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