Generators 50 algebra

Bohm, A., Ne eman, Y., and Barut, A. O. (1988), Dynamical Groups and Spectrum Generating Algebras, World Scientific, Singapore. 

Dothan, Y. (1970), Finite-Dimensional Spectrum-Generating Algebras, Phys. Rev. D 2, 2944. 

Corollary. Let A be a finitely generated algebra over a field. Let T be the set of maximal ideals. Then Spec A is connected iff its subset T is connected. 

Proof. As k[G] is a finitely generated algebra, / is a finitely generated ideal, and I/I2 a finite-dimensional fc-space. By (11.2d) we know the e-derivations A - k are the dual of this space. The first two assertions then are immediate, and the last follows from (11-3) and (11.6). ... 

Show the group map F(k) - G(k) is surjective, pf A B is a finitely generated Hopf subalgebra, AA is Hopf and so faithfully flat over A. Take g A- k, extend its kernel to a maximal ideal of AA , show the residue field is k because a finitely generated algebra. Consider a maximal extension of g to a Hopf subalgebra.]... 

Theorem, (a) Let 0 R be a finitely generated algebra over a field k. Then R has a k-algebra homomorphism to the algebraic closure ... 

This equation was solved for the hydrogen atom by Vladimir Fock [2,3]. The solution in p space revealed the four-dimensional symmetry responsible for the degeneracy of states with the same n but different I quantum numbers in the hydrogen atom. This is a fine example where the momentum-space perspective led to fresh and deep insight. Fock s work spawned much further research on dynamical groups and spectrum-generating algebras. 

The operators J and J 2 are constant within a given irreducible representation of the spectrum-generating algebra they are not of interest here. The effective coupling is given by the scalar product which... 

To begin with, we generalize the spectrum-generating algebra (3.1), introduced for two interacting oscillators, by considering, for N interacting oscillators, the product... 

Once again, nonequispaced data pairs are available for the function f x) at three values of the independent variable xo, xi, and X2, where xi — xq = h and X2 — xi = j (i.e, h j). The objective of this summary of numerical analysis is to generate expressions for d f/dx at x = xo and x = X2. Whereas the previous exercise in numerical analysis in Section 23-3.5 generated central difference results at X = Xi, the discussion in this section focuses on forward and backward differences (see Cutlip and Shacham, 1999, pp. 396-397). These results are employed to generate algebraic difference equations at the lateral boundaries of the control volume. 

Wlien resonances, or near resonances, are present in the 4WM process, the ordering of the field actions in the perturbative treatment (equation (Bl.3.1)). can be highly significant. Though the tliree-coloiir generators (E.s e, E e.s, . . . ) have identical frequency and wavevector algebra, their associated susceptibility... 

By now it should be clear that this kind of operator algebra can be a useful method for generating integrators. We show, in the following, how it can be applied to generate a wide variety of methods for treating the multiple time scale problem. 

For more than three components extremely heavy algebra is generated in attempting to solve the implicit flux relations, and in general no usefully compact explicit solution is obtained. However, there are two interesting special cases in which explicit flux relations can be obtained with an arbitrary nutr er of components in the mixture. Neither would be expected to correspond accurately with physical situations of practical interest, but they may provide useful qualitative, or semi-quantitative pointers to the behavior of more accurate models. 

---