General form of the deformation of Poisson superbracket on (2,2)-dimensional superspace

Continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on n-dimensional space taking values in a Grassmann algebra with m generating elements are described up to an equivalence transformation for the case n=m=2. It is shown that in this case the Poisson superalgebra has an additional deformation comparing with other superdimensions (n,m).Comment: LaTex, 13 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

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

Superconformal Primary Fields on a Graded Riemann Sphere

Primary superfields for a two dimensional Euclidean superconformal field theory are constructed as sections of a sheaf over a graded Riemann sphere. The construction is then applied to the N=3 Neveu-Schwarz case. Various quantities in the N=3 theory are calculated and discussed, such as formal elements of the super-Mobius group, and the two-point function.Comment: LaTeX2e, 23 pages; fixed typos, sorted references, modified definition of primary superfield on page

Cohomologies of the Poisson superalgebra

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $R^{2n}$ ($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

Irreducible holonomy algebras of Riemannian supermanifolds

Possible irreducible holonomy algebras \g\subset\osp(p,q|2m) of Riemannian supermanifolds under the assumption that \g is a direct sum of simple Lie superalgebras of classical type and possibly of a one-dimensional center are classified. This generalizes the classical result of Marcel Berger about the classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.Comment: 27 pages, the final versio

On Alternative Supermatrix Reduction

We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the generalized noninvertible superconformal-like transformations. The features of even- and odd-reduced supermatrices are investigated on a par. They can be unified into some kind of "sandwich" semigroups. Also we define a special module over even- and odd-reduced supermatrix sets, and the generalized Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced supermatrices represent semigroup bands and Rees matrix semigroups over a unit group.Comment: 22 pages, Standard LaTeX with AmS font

Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced.Comment: 25 pages, LaTe

Minimal deformations of the commutative algebra and the linear group GL(n)

We consider the relations of generalized commutativity in the algebra of formal series $M_q (x^i )$, which conserve a tensor $I_q$-grading and depend on parameters $q(i,k)$ . We choose the $I_q$-preserving version of differential calculus on $M_q$ . A new construction of the symmetrized tensor product for $M_q$-type algebras and the corresponding definition of minimally deformed linear group $QGL(n)$ and Lie algebra $qgl(n)$ are proposed. We study the connection of $QGL(n)$ and $qgl(n)$ with the special matrix algebra \mbox{Mat} (n,Q) containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra \mbox{Mat} (n,Q) is given. The exponential parametrization in the algebra \mbox{Mat} (n,Q) is considered on the basis of Campbell-Hausdorf formula.Comment: 14 page

Attitudes of Germans towards distributive issues in the German health system

Social health care systems are inevitably confronted with the scarcity of resources and the resulting distributional challenges. Since prioritization implies distributional effects, decisions on respective rules should take citizens’ preferences into account. Thus, knowledge about citizens’ attitudes and preferences regarding different distributional issues implied by the type of financing health care is necessary to judge the public acceptance of a health system. In this study we concentrate on two distributive issues in the German health system: First, we analyse the acceptance of prioritizing decisions concerning the treatment of certain patient groups, in this case patients who all need a heart operation. Here we focus on the fact that a patient is strong smoker or a non-smoker, the criteria of age or the fact that a patient has or does not have young children. Second, we investigate Germans’ opinions towards income dependent health services. The results reveal strong effects of individuals’ attitudes regarding general aspects of the health system on priorities, e.g. that individuals behaving health demanding should not be preferred. In addition, experiences of limited access to health services are found to have a strong influence on citizens’ attitudes, too. Finally, decisions about different prioritization criteria are found to be not independent.