1 | This is done mostly for notational convenience. If there were other dilatons among the 0-forms, these should be separated
off from the ![]() |
|
2 | Note that we have for convenience chosen to work with a coordinate coframe ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
|
3 | The Hamiltonian heuristic derivation of [48![]() ![]() ![]() |
|
4 | This Hamiltonian exists if ![]() |
|
5 | In this article we will exclusively restrict ourselves to considerations involving the sharp wall limit. However, in recent work
[40![]() |
|
6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
|
7 | Note that the discussion in Footnote 6 applies also here. | |
8 | Note that in the case of the infinite dihedral group ![]() ![]() ![]() ![]() ![]() ![]() |
|
9 | We are employing the convention of Kac [116![]() ![]() |
|
10 | We recall that an ideal ![]() ![]() |
|
11 | Imaginary roots may have arbitrarily negative length squared in general. | |
12 | The generalized Casimir operator ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
|
13 | We discuss in detail a different kind of level decomposition in Section 8. | |
14 | If they were not, one would have by the second point above ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
|
15 | We thank Axel Kleinschmidt for an informative comment on this point. | |
16 | Taking the first spatial direction as compactification direction is convenient, for it does not change the conventions on the
simple roots. More precisely, the Kaluza–Klein ansatz is compatible in that case with our Iwasawa decomposition (2.8![]() ![]() ![]() |
|
17 | This structure of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() For more information about ![]() ![]() |
|
18 | In the following we write simply ![]() ![]() |
|
19 | Actually, the structure constants are integers and thus allows for defining the arithmetic subgroup ![]() |
|
20 | A conjugation on a complex Lie algebra is an antilinear involution, preserving the Lie algebra structure. | |
21 | This decomposition is just the “standard” decomposition of any ![]() |
|
22 | We say that an object is “unique” when it is unique up to an internal automorphism. | |
23 | For example, for the split form ![]() ![]() ![]() |
|
24 | Quite generally, if ![]() ![]() ![]() ![]() ![]() |
|
25 | An algebra is said to be compact if its group of internal automorphisms is compact in the topological sense. A classic theorem states that a semi-simple algebra is compact if and only if its Killing form is negative definite. | |
26 | In the notation ![]() ![]() ![]() |
|
27 | The coefficients ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
|
28 | A system ![]() ![]() ![]() ![]() |
|
29 | Geometrically, this results from the orthogonality of roots ![]() ![]() ![]() ![]() ![]() |
|
30 | If ![]() |
|
31 | The decomposition of ![]() ![]() |
|
32 | D.P. would like to thank Bengt E.W. Nilsson and Jakob Palmkvist for helpful discussions during the creation of
Figure 47![]() |
|
33 | Since we are, in fact, using conjugate Dynkin labels, these conventions are equivalent to the standard ones if one replaces covariant indices by contravariant ones, and vice-versa. | |
34 | Strictly speaking, the coset space defined in this way should be written as ![]() ![]() |
|
35 | As an example, consider the projection ![]() ![]() ![]() ![]() ![]() ![]() |
|
36 | This does not exclude that other approaches would be successful. That ![]() ![]() |
|
37 | One may also consider a point incidence diagram defined as follows: The nodes of the point incidence diagram are the
points of the geometric configuration. Two nodes are joined by a single bond if and only if there is no straight line connecting
the corresponding points. The point incidence diagrams of the configurations ![]() |
|
38 | A true secant is here defined as a line, say ![]() ![]() ![]() |
|
39 | This was also pointed out in [127]. | |
40 | In [90] they were dealing with a hyperbolic internal space so there was an additional ![]() |
|
41 | We recall that a Hamiltonian path is defined as a path in an undirected graph which intersects each node once and only once. A Hamiltonian cycle is then a Hamiltonian path which also returns to its initial node. | |
42 | When no Coxeter exponent ![]() ![]() |
|
43 | To convince oneself of the validity of this commutation relation, it suffices to check it in a basis where the
(finite-dimensional) matrix ![]() ![]() |
|
44 | The uniqueness derives from the fact that the internal automorphism groups of ![]() ![]() |
http://www.livingreviews.org/lrr-2008-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |