498 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
Cartan matrices and presentations of the exceptional simple Elduque Lie superalgebra
Recently Alberto Elduque listed all simple and graded modulo 2 finite
dimensional Lie algebras and superalgebras whose odd component is the spinor
representation of the orthogonal Lie algebra equal to the even component, and
discovered one exceptional such Lie superalgebra in characteristic 5. For this
Lie superalgebra all inequivalent Cartan matrices (in other words, inequivalent
systems of simple roots) are listed together with defining relations between
analogs of its Chevalley generators.Comment: 5 pages, 1 figure, LaTeX2
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 nonholonomic Riemann and Weyl tensors for flag manifolds
On any manifold, any non-degenerate symmetric 2-form (metric) and any
skew-symmetric (differential) form W can be reduced to a canonical form at any
point, but not in any neighborhood: the respective obstructions being the
Riemannian tensor and dW. The obstructions to flatness (to reducibility to a
canonical form) are well-known for any G-structure, not only for Riemannian or
symplectic structures.
For the manifold with a nonholonomic structure (nonintegrable distribution),
the general notions of flatness and obstructions to it, though of huge interest
(e.g., in supergravity) were not known until recently, though particular cases
were known for more than a century (e.g., any contact structure is ``flat'': it
can always be reduced, locally, to a canonical form).
We give a general definition of the NONHOLONOMIC analogs of the Riemann and
Weyl tensors. With the help of Premet's theorems and a package SuperLie we
calculate these tensors for the particular case of flag varieties associated
with each maximal (and several other) parabolic subalgebra of each simple Lie
algebra. We also compute obstructions to flatness of the G(2)-structure and its
nonholonomic super counterpart.Comment: 27 pages, LaTe
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
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
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
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)
Symplectic geometries on supermanifolds
Extension of symplectic geometry on manifolds to the supersymmetric case is
considered. In the even case it leads to the even symplectic geometry (or,
equivalently, to the geometry on supermanifolds endowed with a non-degenerate
Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is
proven that in the odd case there are two different scalar symplectic
structures (namely, an odd closed differential 2-form and the antibracket)
which can be used for construction of symplectic geometries on supermanifolds.Comment: LaTex, 1o pages, LaTex, changed conten
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