Minkowski superspaces and superstrings as almost real-complex supermanifolds

For the Minkowski superspace and superstrings, we define and compute a circumcised analogue of the Nijenhuis tensor, the obstruction to the integrability of an almost real-complex structure. The Nijenhuis tensor vanishes identically only if the superstring superdimension is 1|1 and, moreover, the superstring is endowed with a contact structure. We also show that all real forms of Grassmann algebras are isomorphic, although they are defined by obviously different anti-involutions.     

New simple modular Lie superalgebras as generalized prolongs

Over algebraically closed fields of characteristic p > 2, —prolongations of simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. We discover several new simple Lie superalgebras, serial and exceptional, including super versions of Brown and Melikyan algebras, and thus corroborate the super analog of the Kostrikin-Shafarevich conjecture. Simple Lie superalgebras with 2 × 2 Cartan matrices are classified.     

SO(4)-symmetry of mechanical systems with 3 degrees of freedom

We answered an old question: does there exist a mechanical system with 3 degrees of freedom, except for the Coulomb system, which has 6 first integrals generating the Lie algebra (4) by means of the Poisson brackets? A system which is not centrally symmetric, but has 6 first integrals generating Lie algebra (4), is presented. It is shown also that not every mechanical system with 3 degrees of freedom has first integrals generating (4).     

On the explicit formulas for zeta functions

In this paper, we study the properties of a large class of zeta functions that arises in geometric analysis and mathematical physics. They are attached to some elliptic operators. This method can be used to evaluate explicitly the special values of zeta functions of elliptic operators defined on some symmetric spaces.     

Projectively Equivariant Quantization Map

Let M be a manifold endowed with a symmetric affine connection . The aim of this Letter is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T* M and the space of second-order linear differential operators, both viewed as modules over the group of diffeomorphisms and the Lie algebra of vector fields on M. This map is an isomorphism, for almost all values of certain constants, and it depends only on the projective class of the affine connection .