260 research outputs found
Gel'fand-Zetlin Basis and Clebsch-Gordan Coefficients for Covariant Representations of the Lie superalgebra gl(m|n)
A Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor
representations of the Lie superalgebra gl(m|n). Explicit expressions for the
generators of the Lie superalgebra acting on this basis are determined.
Furthermore, Clebsch-Gordan coefficients corresponding to the tensor product of
any covariant tensor representation of gl(m|n) with the natural representation
V ([1,0,...,0]) of gl(m|n) with highest weight (1,0,. . . ,0) are computed.
Both results are steps for the explicit construction of the parastatistics Fock
space.Comment: 16 page
General form of the deformation of Poisson superbracket on (2,2)-dimensional superspace
Continuous formal deformations of the Poisson superbracket defined on
compactly supported smooth functions on n-dimensional space taking values in a
Grassmann algebra with m generating elements are described up to an equivalence
transformation for the case n=m=2. It is shown that in this case the Poisson
superalgebra has an additional deformation comparing with other superdimensions
(n,m).Comment: LaTex, 13 page
Cohomologies of the Poisson superalgebra
Cohomology spaces of the Poisson superalgebra realized on smooth
Grassmann-valued functions with compact support on ($C^{2n}) are
investigated under suitable continuity restrictions on cochains. The first and
second cohomology spaces in the trivial representation and the zeroth and first
cohomology spaces in the adjoint representation of the Poisson superalgebra are
found for the case of a constant nondegenerate Poisson superbracket for
arbitrary n>0. The third cohomology space in the trivial representation and the
second cohomology space in the adjoint representation of this superalgebra are
found for arbitrary n>1.Comment: Comments: 40 pages, the text to appear in Theor. Math. Phys.
supplemented by computation of the 3-rd trivial cohomolog
Realizations of the Lie superalgebra q(2) and applications
The Lie superalgebra q(2) and its class of irreducible representations V_p of
dimension 2p (p being a positive integer) are considered. The action of the
q(2) generators on a basis of V_p is given explicitly, and from here two
realizations of q(2) are determined. The q(2) generators are realized as
differential operators in one variable x, and the basis vectors of V_p as
2-arrays of polynomials in x. Following such realizations, it is observed that
the Hamiltonian of certain physical models can be written in terms of the q(2)
generators. In particular, the models given here as an example are the
sphaleron model, the Moszkowski model and the Jaynes-Cummings model. For each
of these, it is shown how the q(2) realization of the Hamiltonian is helpful in
determining the spectrum.Comment: LaTeX file, 15 pages. (further references added, minor changes in
section 5
Centre and Representations of U_q(sl(2|1)) at Roots of Unity
Quantum groups at roots of unity have the property that their centre is
enlarged. Polynomial equations relate the standard deformed Casimir operators
and the new central elements. These relations are important from a physical
point of view since they correspond to relations among quantum expectation
values of observables that have to be satisfied on all physical states. In this
paper, we establish these relations in the case of the quantum Lie superalgebra
U_q(sl(2|1)). In the course of the argument, we find and use a set of
representations such that any relation satisfied on all the representations of
the set is true in U_q(sl(2|1)). This set is a subset of the set of all the
finite dimensional irreducible representations of U_q(sl(2|1)), that we
classify and describe explicitly.Comment: Minor corrections, References added. LaTeX2e, 18 pages, also
available at http://lapphp0.in2p3.fr/preplapp/psth/ENSLAPP583.ps.gz . To
appear in J. Phys. A: Math. Ge
Wigner quantum oscillators. Osp(3/2) oscillators
The properties of the three-dimensional noncanonical osp(3/2) oscillators,
introduced in J.Phys. A: Math. Gen. {\bf 27} (1994) 977, are further studied.
The angular momentum M of the oscillators can take at most three values
M=p-1,p,p+1, which are either all integers or all half-integers. The
coordinates anticommute with each other. Depending on the state space the
energy spectrum can coincide or can be essentially different from those of the
canonical oscillator. The ground state is in general degenerated.Comment: TeX, Preprint INRNE-TH-94/3, 17
Exact exponents for the spin quantum Hall transition
We consider the spin quantum Hall transition which may occur in disordered
superconductors with unbroken SU(2) spin-rotation symmetry but broken
time-reversal symmetry. Using supersymmetry, we map a model for this transition
onto the two-dimensional percolation problem. The anisotropic limit is an
sl(2|1) supersymmetric spin chain. The mapping gives exact values for critical
exponents associated with disorder-averages of several observables in good
agreement with recent numerical results.Comment: 5 pages, 2 figure
Matrix difference equations for the supersymmetric Lie algebra sl(2,1) and the `off-shell' Bethe ansatz
Based on the rational R-matrix of the supersymmetric sl(2,1) matrix
difference equations are solved by means of a generalization of the nested
algebraic Bethe ansatz. These solutions are shown to be of highest-weight with
respect to the underlying graded Lie algebra structure.Comment: 10 pages, LaTex, references and acknowledgements added, spl(2,1) now
called sl(2,1
Finite-dimensional representations of the quantum superalgebra and related q-identities
Explicit expressions for the generators of the quantum superalgebra
acting on a class of irreducible representations are given. The
class under consideration consists of all essentially typical representations:
for these a Gel'fand-Zetlin basis is known. The verification of the quantum
superalgebra relations to be satisfied is shown to reduce to a set of
-number identities.Comment: 12 page
New solutions to the Reflection Equation and the projecting method
New integrable boundary conditions for integrable quantum systems can be
constructed by tuning of scattering phases due to reflection at a boundary and
an adjacent impurity and subsequent projection onto sub-spaces. We illustrate
this mechanism by considering a gl(m<n)-impurity attached to an open
gl(n)-invariant quantum chain and a Kondo spin S coupled to the supersymmetric
t-J model.Comment: Latex2e, no figure
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