269 research outputs found
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 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)
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
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
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
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
Even and odd symplectic and K\"ahlerian structures on projective superspaces
Supergeneralization of \DC P(N) provided by even and odd K\"ahlerian
structures from Hamiltonian reduction are construct.Operator which
used in Batalin-- Vilkovisky quantization formalism and mechanics which are
bi-Hamiltonian under corresponding even and odd Poisson brackets are
considered.Comment: 19 page
Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?
We discuss of the conceptual difficulties connected with the
anticommutativity of classical fermion fields, and we argue that the "space" of
all classical configurations of a model with such fields should be described as
an infinite-dimensional supermanifold M.
We discuss the two main approaches to supermanifolds, and we examine the
reasons why many physicists tend to prefer the Rogers approach although the
Berezin-Kostant-Leites approach is the more fundamental one. We develop the
infinite-dimensional variant of the latter, and we show that the functionals on
classical configurations considered in a previous paper are nothing but
superfunctions on M. We present a programme for future mathematical work, which
applies to any classical field model with fermion fields. This programme is
(partially) implemented in successor papers.Comment: 46 pages, LateX2E+AMSLaTe
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
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