On bilinear invariant differential operators acting on tensor fields on the symplectic manifold

Let $M$ be an $n$-dimensional manifold, $V$ the space of a representation $\rho: GL(n)\longrightarrow GL(V)$. Locally, let $T(V)$ be the space of sections of the tensor bundle with fiber $V$ over a sufficiently small open set $U\subset M$, in other words, $T(V)$ is the space of tensor fields of type $V$ on $M$ on which the group \Diff (M) of diffeomorphisms of $M$ naturally acts. Elsewhere, the author classified the \Diff (M)-invariant differential operators $D: T(V_{1})\otimes T(V_{2})\longrightarrow T(V_{3})$ 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 $(M, \omega)$. 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 $M$

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

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

Gauge invariant formulation of N=2N=2 Toda and KdV systems in extended superspace

We give a gauge invariant formulation of $N=2$ 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

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)

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

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

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