244 research outputs found
The second cohomology of sl(m|1) with coefficients in its enveloping algebra is trivial
Using techniques developed in a recent article by the authors, it is proved
that the 2-cohomology of the Lie superalgebra sl(m|1); m > 1, with coefficients
in its enveloping algebra is trivial. The obstacles in solving the analogous
problem for sl(3|2) are also discussed.Comment: 15 pages, Latex, no figure
Cohomology of Lie superalgebras and of their generalizations
The cohomology groups of Lie superalgebras and, more generally, of color Lie
algebras, are introduced and investigated. The main emphasis is on the case
where the module of coefficients is non-trivial. Two general propositions are
proved, which help to calculate the cohomology groups. Several examples are
included to show the peculiarities of the super case. For L = sl(1|2), the
cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple
graded L-module, are determined, and the result is used to show that
H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies
that the superalgebra U(L) does not admit of any non-trivial formal
deformations (in the sense of Gerstenhaber). Garland's theory of universal
central extensions of Lie algebras is generalized to the case of color Lie
algebras.Comment: 50 pages, Latex, no figures. In the revised version the proof of
Lemma 5.1 is greatly simplified, some references are added, and a pertinent
result on sl(m|1) is announced. To appear in the Journal of Mathematical
Physic
A super-analogue of Kontsevich's theorem on graph homology
In this paper we will prove a super-analogue of a well-known result by
Kontsevich which states that the homology of a certain complex which is
generated by isomorphism classes of oriented graphs can be calculated as the
Lie algebra homology of an infinite-dimensional Lie algebra of symplectic
vector fields.Comment: 15 page
Finite dimensional representations of at arbitrary
A method is developed to construct irreducible representations(irreps) of the
quantum supergroup in a systematic fashion. It is shown that
every finite dimensional irrep of this quantum supergroup at generic is a
deformation of a finite dimensional irrep of its underlying Lie superalgebra
, and is essentially uniquely characterized by a highest weight. The
character of the irrep is given. When is a root of unity, all irreps of
are finite dimensional; multiply atypical highest weight irreps
and (semi)cyclic irreps also exist. As examples, all the highest weight and
(semi)cyclic irreps of are thoroughly studied.Comment: 21 page
Type-I Quantum Superalgebras, -Supertrace and Two-variable Link Polynomials
A new general eigenvalue formula for the eigenvalues of Casimir invariants,
for the type-I quantum superalgebras, is applied to the construction of link
polynomials associated with {\em any} finite dimensional unitary irrep for
these algebras. This affords a systematic construction of new two-variable link
polynomials asociated with any finite dimensional irrep (with a real highest
weight) for the type-I quantum superalgebras. In particular infinite families
of non-equivalent two-variable link polynomials are determined in fully
explicit form.Comment: the version to be published in J. Math. Phy
Tensor operators and Wigner-Eckart theorem for the quantum superalgebra U_{q}[osp(1\mid 2)]
Tensor operators in graded representations of Z_{2}-graded Hopf algebras are
defined and their elementary properties are derived. Wigner-Eckart theorem for
irreducible tensor operators for U_{q}[osp(1\mid 2)] is proven. Examples of
tensor operators in the irreducible representation space of Hopf algebra
U_{q}[osp(1\mid 2)] are considered. The reduced matrix elements for the
irreducible tensor operators are calculated. A construction of some elements of
the center of U_{q}[osp(1\mid 2)] is given.Comment: 16 pages, Late
On representations of super coalgebras
The general structure of the representation theory of a -graded
coalgebra is discussed. The result contains the structure of Fourier analysis
on compact supergroups and quantisations thereof as a special case. The general
linear supergroups serve as an explicit illustration and the simplest example
is carried out in detail.Comment: 18 pages, LaTeX, KCL-TH-94-
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
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