391 research outputs found
On bilinear invariant differential operators acting on tensor fields on the symplectic manifold
Let be an -dimensional manifold, the space of a representation
. Locally, let be the space of
sections of the tensor bundle with fiber over a sufficiently small open set
, in other words, is the space of tensor fields of type
on on which the group \Diff (M) of diffeomorphisms of naturally acts.
Elsewhere, the author classified the \Diff (M)-invariant differential
operators for irreducible
fibers with lowest weight. Here the result is generalized to bilinear operators
invariant with respect to the group \Diff_{\omega}(M) of symplectomorphisms
of the symplectic manifold . We classify all first order invariant
operators; the list of other operators is conjectural. Among the new operators
we mention a 2nd order one which determins an ``algebra'' structure on the
space of metrics (symmetric forms) on
On Einstein equations on manifolds and supermanifolds
The Einstein equations (EE) are certain conditions on the Riemann tensor on
the real Minkowski space M. In the twistor picture, after complexification and
compactification M becomes the Grassmannian of 2-dimensional
subspaces in the 4-dimensional complex one. Here we answer for which of the
classical domains considered as manifolds with G-structure it is possible to
impose conditions similar in some sense to EE. The above investigation has its
counterpart on superdomains: an analog of the Riemann tensor is defined for any
supermanifold with G-structure with any Lie supergroup G. We also derive
similar analogues of EE on supermanifolds. Our analogs of EE are not what
physicists consider as SUGRA (supergravity), for SUGRA see \cite{GL4,LP2}.Comment: arxiv version is already officia
Hyperbolic Kac-Moody superalgebras
We present a classification of the hyperbolic Kac-Moody (HKM) superalgebras.
The HKM superalgebras of rank larger or equal than 3 are finite in number (213)
and limited in rank (6). The Dynkin-Kac diagrams and the corresponding simple
root systems are determined. We also discuss a class of singular
sub(super)algebras obtained by a folding procedure
The Binary Invariant Differential Operators on Weighted Densities on the superspace and Cohomology
Over the -dimensional real superspace, , we classify
-invariant binary differential operators acting on the
superspaces of weighted densities, where is the Lie
superalgebra of contact vector fields. This result allows us to compute the
first differential cohomology of %the Lie superalgebra with
coefficients in the superspace of linear differential operators acting on the
superspaces of weighted densities--a superisation of a result by Feigin and
Fuchs. We explicitly give 1-cocycles spanning these cohomology spaces
The anticommutator spin algebra, its representations and quantum group invariance
We define a 3-generator algebra obtained by replacing the commutators by
anticommutators in the defining relations of the angular momentum algebra. We
show that integer spin representations are in one to one correspondence with
those of the angular momentum algebra. The half-integer spin representations,
on the other hand, split into two representations of dimension j + 1/2. The
anticommutator spin algebra is invariant under the action of the quantum group
SO_q(3) with q=-1.Comment: 7 A4 page
Gauge invariant formulation of Toda and KdV systems in extended superspace
We give a gauge invariant formulation of supersymmetric abelian Toda
field equations in \n2 superspace. Superconformal invariance is studied. The
conserved currents are shown to be associated with Drinfeld-Sokolov type
gauges. The extension to non-abelian \n2 Toda equations is discussed. Very
similar methods are then applied to a matrix formulation in \n2 superspace of
one of the \n2 KdV hierarchies.Comment: 21 page
Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them
Among the simple finite dimensional Lie algebras, only sl(n) possesses two
automorphisms of finite order which have no common nonzero eigenvector with
eigenvalue one. It turns out that these automorphisms are inner and form a pair
of generators that allow one to generate all of sl(n) under bracketing. It
seems that Sylvester was the first to mention these generators, but he used
them as generators of the associative algebra of all n times n matrices Mat(n).
These generators appear in the description of elliptic solutions of the
classical Yang-Baxter equation, orthogonal decompositions of Lie algebras, 't
Hooft's work on confinement operators in QCD, and various other instances. Here
I give an algorithm which both generates sl(n) and explicitly describes a set
of defining relations. For simple (up to center) Lie superalgebras, analogs of
Sylvester generators exist only for sl(n|n). The relations for this case are
also computed.Comment: 14 pages, 6 figure
The Shapovalov determinant for the Poisson superalgebras
Among simple Z-graded Lie superalgebras of polynomial growth, there are
several which have no Cartan matrix but, nevertheless, have a quadratic even
Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields
on the (1|6)-dimensional supercircle preserving the contact form, and the
series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian
fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using
C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for
the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov
described irreducible finite dimensional representations of po(0|n) and
sh(0|n); we generalize his result for Verma modules: give criteria for
irreducibility of the Verma modules over po(0|2k) and sh(0|2k)
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
Minkowski superspaces and superstrings as almost real-complex supermanifolds
In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that
mathematicians study are real or (almost) complex ones, while Minkowski
superspaces are completely different objects. They are what we call almost
real-complex supermanifolds, i.e., real supermanifolds with a non-integrable
distribution, the collection of subspaces of the tangent space, and in every
subspace a complex structure is given.
An almost complex structure on a real supermanifold can be given by an even
or odd operator; it is complex (without "always") if the suitable superization
of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we
define the circumcised analog of the Nijenhuis tensor. We compute it for the
Minkowski superspaces and superstrings. The space of values of the circumcised
Nijenhuis tensor splits into (indecomposable, generally) components whose
irreducible constituents are similar to those of Riemann or Penrose tensors.
The Nijenhuis tensor vanishes identically only on superstrings of
superdimension 1|1 and, besides, the superstring is endowed with a contact
structure. We also prove that all real forms of complex Grassmann algebras are
isomorphic although singled out by manifestly different anti-involutions.Comment: Exposition of the same results as in v.1 is more lucid. Reference to
related recent work by Witten is adde
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