6,732 research outputs found
Hidden supersymmetries in supersymmetric quantum mechanics
We discuss the appearance of additional, hidden supersymmetries for simple
0+1 -invariant supersymmetric models and analyse some geometrical
mechanisms that lead to them. It is shown that their existence depends
crucially on the availability of odd order invariant skewsymmetric tensors on
the (generic) compact Lie algebra , and hence on the cohomology
properties of the Lie algebra considered.Comment: Misprints corrected, two refs. added. To appear in NP
Optimally defined Racah-Casimir operators for su(n) and their eigenvalues for various classes of representations
This paper deals with the striking fact that there is an essentially
canonical path from the -th Lie algebra cohomology cocycle, ,
of a simple compact Lie algebra \g of rank to the definition of its
primitive Casimir operators of order . Thus one obtains a
complete set of Racah-Casimir operators for each \g and nothing
else. The paper then goes on to develop a general formula for the eigenvalue
of each valid for any representation of \g, and thereby
to relate to a suitably defined generalised Dynkin index. The form of
the formula for for is known sufficiently explicitly to make
clear some interesting and important features. For the purposes of
illustration, detailed results are displayed for some classes of representation
of , including all the fundamental ones and the adjoint representation.Comment: Latex, 16 page
Group Theoretical Foundations of Fractional Supersymmetry
Fractional supersymmetry denotes a generalisation of supersymmetry which may
be constructed using a single real generalised Grassmann variable, , for arbitrary integer . An
explicit formula is given in the case of general for the transformations
that leave the theory invariant, and it is shown that these transformations
possess interesting group properties. It is shown also that the two generalised
derivatives that enter the theory have a geometric interpretation as generators
of left and right transformations of the fractional supersymmetry group.
Careful attention is paid to some technically important issues, including
differentiation, that arise as a result of the peculiar nature of quantities
such as .Comment: Plain Latex, 18 page
Compilation of relations for the antisymmetric tensors defined by the Lie algebra cocycles of
This paper attempts to provide a comprehensive compilation of results, many
new here, involving the invariant totally antisymmetric tensors (Omega tensors)
which define the Lie algebra cohomology cocycles of , and that play an
essential role in the optimal definition of Racah-Casimir operators of .
Since the Omega tensors occur naturally within the algebra of totally
antisymmetrised products of -matrices of , relations within
this algebra are studied in detail, and then employed to provide a powerful
means of deriving important Omega tensor/cocycle identities. The results
include formulas for the squares of all the Omega tensors of . Various
key derivations are given to illustrate the methods employed.Comment: Latex file (run thrice). Misprints corrected, Refs. updated.
Published in IJMPA 16, 1377-1405 (2001
Effective actions, relative cohomology and Chern Simons forms
The explicit expression of all the WZW effective actions for a simple group G
broken down to a subgroup H is established in a simple and direct way, and the
formal similarity of these actions to the Chern-Simons forms is explained.
Applications are also discussed.Comment: 11 pages. Latex2e file. Published versio
Prevalence of five common clinical abnormalities in very elderly people: population based cross sectional study
As the prevalence of disease rises with age, the number of people with unidentified abnormalities is also likely to increase. We assessed the number of previously known and newly identified patients with anaemia, diabetes mellitus, thyroid dysfunction, atrial fibrillation, and hypertension in a population based sample of 85 year old people
Invariant tensors for simple groups
The forms of the invariant primitive tensors for the simple Lie algebras A_l,
B_l, C_l and D_l are investigated. A new family of symmetric invariant tensors
is introduced using the non-trivial cocycles for the Lie algebra cohomology.
For the A_l algebra it is explicitly shown that the generic forms of these
tensors become zero except for the l primitive ones and that they give rise to
the l primitive Casimir operators. Some recurrence and duality relations are
given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3)
and su(4) are also provided. Finally, new relations involving the d and f su(n)
tensors are given.Comment: Latex file. 34 pages. (Trivial) misprints corrected. To appear in
Nucl. Phys.
Geometrical foundations of fractional supersymmetry
A deformed -calculus is developed on the basis of an algebraic structure
involving graded brackets. A number operator and left and right shift operators
are constructed for this algebra, and the whole structure is related to the
algebra of a -deformed boson. The limit of this algebra when is a -th
root of unity is also studied in detail. By means of a chain rule expansion,
the left and right derivatives are identified with the charge and covariant
derivative encountered in ordinary/fractional supersymmetry and this leads
to new results for these operators. A generalized Berezin integral and
fractional superspace measure arise as a natural part of our formalism. When
is a root of unity the algebra is found to have a non-trivial Hopf
structure, extending that associated with the anyonic line. One-dimensional
ordinary/fractional superspace is identified with the braided line when is
a root of unity, so that one-dimensional ordinary/fractional supersymmetry can
be viewed as invariance under translation along this line. In our construction
of fractional supersymmetry the -deformed bosons play a role exactly
analogous to that of the fermions in the familiar supersymmetric case.Comment: 42 pages, LaTeX. To appear in Int. J. Mod. Phys.
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