274 research outputs found
An Exactly Solvable Spin Chain Related to Hahn Polynomials
We study a linear spin chain which was originally introduced by Shi et al.
[Phys. Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength
contains a parameter and depends on the parity of the chain site.
Extending the model by a second parameter , it is shown that the single
fermion eigenstates of the Hamiltonian can be computed in explicit form. The
components of these eigenvectors turn out to be Hahn polynomials with
parameters and . The construction of the
eigenvectors relies on two new difference equations for Hahn polynomials. The
explicit knowledge of the eigenstates leads to a closed form expression for the
correlation function of the spin chain. We also discuss some aspects of a
-extension of this model
A class of infinite-dimensional representations of the Lie superalgebra osp(2m+1|2n) and the parastatistics Fock space
An orthogonal basis of weight vectors for a class of infinite-dimensional
representations of the orthosymplectic Lie superalgebra osp(2m+1|2n) is
introduced. These representations are particular lowest weight representations
V(p), with a lowest weight of the form [-p/2,...,-p/2|p/2,...,p/2], p being a
positive integer. Explicit expressions for the transformation of the basis
under the action of algebra generators are found. Since the relations of
algebra generators correspond to the defining relations of m pairs of
parafermion operators and n pairs of paraboson operators with relative
parafermion relations, the parastatistics Fock space of order p is also
explicitly constructed. Furthermore, the representations V(p) are shown to have
interesting characters in terms of supersymmetric Schur functions, and a simple
character formula is also obtained.Comment: 15 page
A classification of generalized quantum statistics associated with basic classical Lie superalgebras
Generalized quantum statistics such as para-statistics is usually
characterized by certain triple relations. In the case of para-Fermi statistics
these relations can be associated with the orthogonal Lie algebra B_n=so(2n+1);
in the case of para-Bose statistics they are associated with the Lie
superalgebra B(0|n)=osp(1|2n). In a previous paper, a mathematical definition
of ``a generalized quantum statistics associated with a classical Lie algebra
G'' was given, and a complete classification was obtained. Here, we consider
the definition of ``a generalized quantum statistics associated with a basic
classical Lie superalgebra G''. Just as in the Lie algebra case, this
definition is closely related to a certain Z-grading of G. We give in this
paper a complete classification of all generalized quantum statistics
associated with the basic classical Lie superalgebras A(m|n), B(m|n), C(n) and
D(m|n)
Gel'fand-Zetlin basis for a class of representations of the Lie superalgebra gl(\infty|\infty)
A new, so called odd Gel'fand-Zetlin basis is introduced for the irreducible
covariant tensor representations of the Lie superalgebra gl(n|n). The related
Gel'fand-Zetlin patterns are based upon the decomposition according to a
particular chain of subalgebras of gl(n|n). This chain contains only genuine
Lie superalgebras of type gl(k|l) with k and l nonzero (apart from the final
element of the chain which is gl(1|0)=gl(1)). Explicit expressions for a set of
generators of the algebra on this Gel'fand-Zetlin basis are determined. The
results are extended to an explicit construction of a class of irreducible
highest weight modules of the general linear Lie superalgebra
gl(\infty|\infty).Comment: 21 page
Solutions of the compatibility conditions for a Wigner quantum oscillator
We consider the compatibility conditions for a N-particle D-dimensional
Wigner quantum oscillator. These conditions can be rewritten as certain triple
relations involving anticommutators, so it is natural to look for solutions in
terms of Lie superalgebras. In the recent classification of ``generalized
quantum statistics'' for the basic classical Lie superalgebras
[math-ph/0504013], each such statistics is characterized by a set of creation
and annihilation operators plus a set of triple relations. In the present
letter, we investigate which cases of this classification also lead to
solutions of the compatibility conditions. Our analysis yields some known
solutions and several classes of new solutions.Comment: 9 page
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