161 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
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
Biconformal supergravity and the AdS/CFT conjecture
Biconformal supergravity models provide a new gauging of the superconformal
group relevant to the Maldacena conjecture. Using the group quotient method to
biconformally gauge SU(2,2|N), we generate a 16-dim superspace. We write the
most general even- and odd-parity actions linear in the curvatures, the bosonic
sector of which is known to descend to general relativity on a 4-dim manifold.Comment: 35 pages, adjusted group nomenclature, 1 reference and
acknowledgements adde
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
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
Strings from Gauged Wess-Zumino-Witten Models
We present an algebraic approach to string theory. An embedding of
in a super Lie algebra together with a grading on the Lie algebra determines a
nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra
in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of
the Wess-Zumino-Witten model to some extension of the superconformal
algebra. The extension is completely determined by the embedding. The
realization of the superconformal algebra is determined by the grading. For a
particular choice of grading, one obtains in this way, after twisting, the BRST
structure of a string theory. We classify all embeddings of into Lie
super algebras and give a detailed account of the branching of the adjoint
representation. This provides an exhaustive classification and characterization
of both all extended superconformal algebras and all string theories
which can be obtained in this way.Comment: 50 pages, LaTe
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