172 research outputs found
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
A classification of generalized quantum statistics associated with classical Lie algebras
Generalized quantum statistics such as para-Fermi statistics is characterized
by certain triple relations which, in the case of para-Fermi statistics, are
related to the orthogonal Lie algebra B_n=so(2n+1). In this paper, we give a
quite general definition of ``a generalized quantum statistics associated to a
classical Lie algebra G''. This definition is closely related to a certain
Z-grading of G. The generalized quantum statistics is then determined by a set
of root vectors (the creation and annihilation operators of the statistics) and
the set of algebraic relations for these operators. Then we give a complete
classification of all generalized quantum statistics associated to the
classical Lie algebras A_n, B_n, C_n and D_n. In the classification, several
new classes of generalized quantum statistics are described
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