Coset space

We begin with the simple case of one-dimensional problems described algebraically by U(2). The coset space for this case is just a single complex variable, which we call We denote the complex conjugate by 2,. These variables can be interpreted in terms of the position (q) and momentum (p) variables in phase space. Equivalently the t, variables can be related to the action-angle variables /,0 introduced in Section 3.4. To be more precise... 

The method discussed in Sections 7.5-7.7 is particularly useful for coupled problems. We begin the discussion by considering two coupled onedimensional degrees of freedom described algebraically by U ) ) (Section 4.2). The coset space is here composed of two complex variables, and i 2, describing the coordinates and momenta of the two bonds... 

The algebraic description of three-dimensional problems is in terms of U(4). The coset space for this case is a complex three-dimensional vector, We denote its complex conjugate by q. One can then introduce the canonical position and momentum variables q and p by the transformation... 

We consider next the case of two three-dimensional coupled problems described by U(1)(4) U<2,(4). The coset space is here composed of two com-... 

The concept of a coset space is discussed in detail in books on group theory (Gilmore, 1974) and is reviewed in Chapter 3 of Iachello and Arima (1987). The coset spaces of interest for algebraic models with structure U(n) are the spaces U(n)/U(n - 1) U(l). These spaces are complex spaces with (n - 1) complex variables (coordinates and momenta). 

In other words, we can obtain a unique coset space for a given R). 

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. 

What really needs to be done here is to expand the whole framework to include non-affine schemes (3.6). The projective line is such a scheme, covered by two overlapping copies of the ordinary line and in fact one can always get coset spaces as schemes. Indeed, we have already seen part of the proof. If say H s G are algebraic matrix groups, there is some V st k" with G-action where H is the stabilizer of a one-dimensional subspace, and this matches up the H-cosets with other such subspaces, points in projective (n — l)-space. But even if we had the general result, it would take substantial extra work to show that for normal subgroups the coset space is affine. We will just give a direct proof of this case. 

It will be useful to have in mind another way of considering the problem a function on a coset space of G is essentially a function on G invariant under translation by the subgroup. When G is GL and H the upper triangular group, for instance, it is easy to compute that no nonconstant polynomial in the matrix entries is invariant under all translations by elements of H, and thus no affine coset space can exist. (What follows from (16.1) is that there are always semi-invariant functions, ones where each translate of/is a constant multiple of/) Our problem is to prove the existence of a large collection of invariant functions for normal subgroups. 

Correspondingly, in the theory of the chemical identity groups stereochemical features of the molecules and EMs are represented by their chemical identity groups and their left cosets in SymL, and the permutational isomerizations and the stereochemical aspect of chemical reactions are described by the so-called set-valued mappings of the left coset spaces of the respective chemical identity groups [10, 19, 35]. 

Finally, reductive groups play a major role in recent work on automor-phic functions. To take the basic example, let k be the reals. Then SL2(k) acts on the half-plane 2 = x + iy y > 0 by ( J)z = (az + b)/(cz + d) this is transitive, and the circle group K - ( g ) a2 + b2 = 1 is the stabilizer of z = i. Thus the half-plane is the coset space (symmetric space) for K in SLz(k). The classical modular functions on the half-plane are precisely those invariant under the arithmetic subgroup SL2(Z) or certain subgroups of it. All such functions can thus be pulled back to be functions on SL2(k) with certain invariance properties. The same thing then can be done for coset spaces of other reductive groups. Some of the most recent treatments also use the group not just for the reals but for the various p-adic completions of... 

Each element of the coset space G/H then corresponds to a coherent state. The decomposition of the group into cosets, taking advantage of the stability group properties, reduces the parameter space of the coherent state to a nonredundant set. In our case the stability group is SO(2) and we can write... 

The geometrical interpretation is constructed through a two-dimensional coset space spanned by two complex variables... 

This permutational representation is also called the ground representation. It describes the transformation of the coset space. The dimension of this coset space is G / //. In the case of a cluster, where each coset corresponds to a site, it represents the permutation of the positions of the sites. For this reason, it is also called the positional representation. Indeed, Eq. (4.72) may equally well be written as... 

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