8 research outputs found
Generalized MICZ-Kepler Problems and Unitary Highest Weight Modules
For each integer , we demonstrate that a -dimensional
generalized MICZ-Kepler problem has an \mr{Spin}(2, 2n+2) dynamical symmetry
which extends the manifest \mr{Spin}(2n+1) symmetry. The Hilbert space of
bound states is shown to form a unitary highest weight \mr{Spin}(2,
2n+2)-module which occurs at the first reduction point in the
Enright-Howe-Wallach classification diagram for the unitary highest weight
modules. As a byproduct, we get a simple geometric realization for such a
unitary highest weight \mr{Spin}(2, 2n+2)-module.Comment: 27 pages, Refs. update
Can fusion coefficients be calculated from the depth rule ?
The depth rule is a level truncation of tensor product coefficients expected
to be sufficient for the evaluation of fusion coefficients. We reformulate the
depth rule in a precise way, and show how, in principle, it can be used to
calculate fusion coefficients. However, we argue that the computation of the
depth itself, in terms of which the constraints on tensor product coefficients
is formulated, is problematic. Indeed, the elements of the basis of states
convenient for calculating tensor product coefficients do not have a
well-defined depth! We proceed by showing how one can calculate the depth in an
`approximate' way and derive accurate lower bounds for the minimum level at
which a coupling appears. It turns out that this method yields exact results
for and constitutes an efficient and simple algorithm for
computing fusion coefficients.Comment: 27 page
Compact Lie groups and their representations
The content of this book is somewhat different from that of traditional books on representation theory. First, bearing in mind the needs of physicists, the author has tried to make the exposition as elementary as possible. The need for an elementary exposition has influenced the distribution of the material. The book is divided into three largely independent parts, arranged in order of increasing difficulty. Besides compact Lie groups, groups with other topological structure ("similar" to compact groups in some sense) are considered. Prominent among these are reductive complex Lie groups (including semisimple groups), obtained from compact Lie groups by analytic continuation, and also their real forms (reductive real Lie groups). The theory of finite-dimensional representation for these classes of groups is developed, striving whenever possible to emphasize the "compact origin" of these representations, i.e., their analytic relationship to representations of compact Lie groups. In conclusion, the author briefly presents some aspects of infinite-dimensional representations of semisimple complex Lie algebras and Lie groups