Communications in Mathematics |

Editors: Dimitry Leites & Sofiane Bouarroudj

Selected stories about the life of A. L. Onishchik, and a review of hiscontribution to the classification of non-split supermanifolds, in particular,supercurves a.k.a. superstrings; his editorial and educational work. A briefoverview of his and his students' results in supersymmetry, and their impact onother researchers. Several open problems growing out of Onishchik's research are presented, someof them are related with odd parameters of deformations and non-holonomicstructures of supermanifolds important in physical models, such as Minkowskisuperspaces and certain superstrings.

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheafof sections of the exterior algebra of the homogeneous vector bundle $E$ overthe flag variety $G/P$, where $G$ is a simple finite-dimensional complex Liegroup and $P$ its parabolic subgroup. Then, $\mathfrak{d}$ is transitive andirreducible whenever $E$ is defined by an irreducible $P$-module $V$ such thatthe highest weight of $V^*$ is dominant. Moreover, $\mathfrak{d}$ is simple; itis isomorphic to the Lie superalgebra of vector fields on the superpoint, i.e.,on a $0|n$-dimensional supervariety.

The "curved" super Grassmannian is the supervariety of subsupervarieties ofpurely odd dimension $k$ in a~supervariety of purely odd dimension $n$, unlikethe "usual" super Grassmannian which is the supervariety of linearsubsuperspacies of purely odd dimension $k$ in a~superspace of purely odddimension $n$. The Lie superalgebras of all and Hamiltonian vector fields onthe superpoint are realized as Lie superalgebras of derivations of thestructure sheaves of certain "curved" super Grassmannians,

Here, I study the problem of classification of non-split supermanifoldshaving as retract the split supermanifold $(M,\Omega)$, where $\Omega$ is thesheaf of holomorphic forms on a given complex manifold $M$ of dimension $> 1$.I propose a general construction associating with any $d$-closed $(1,1)$-form$\omega$ on $M$ a supermanifold with retract $(M,\Omega)$ which is non-splitwhenever the Dolbeault class of $\omega$ is non-zero. In particular, this givesa non-empty family of non-split supermanifolds for any flag manifold $M\ne\mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitiansymmetric space, I get a complete classification of non-split supermanifoldswith retract $(M,\Omega)$. For each of these supermanifolds, the 0- and1-cohomology with values in the tangent sheaf are calculated. As an example, Istudy the $\Pi$-symmetric super-Grassmannians introduced by Yu. Manin.

I classified bilinear differential operators acting in the spaces of tensorfields on any real or complex manifold and invariant with respect to thediffeomorphisms in 1980. Here I give the details of the proof.

We describe functorially the first Galois cohomology set $H^1({\mathbb R},G)$of a connected reductive algebraic group $G$ over the field $\mathbb R$ of realnumbers in terms of a certain action of the Weyl group on the real points oforder dividing 2 of the maximal torus containing a maximal compact torus. Thisresult was announced with a sketch of proof in the author's 1988 note. Here wegive a detailed proof.

In this note I prove a~claim on determinants of some special tridiagonalmatrices. Together with my result about Fibonacci partitions(arXiv:math/0307150), this claim allows one to prove one (slightlystrengthened) Shallit's result about such partitions.

Any complex-analytic supermanifold whose retract is diffeomorphic to the complex projective superline (superstring) $CP^{1|4}$ is, up to a diffeomorphism, either a member of a 1-parameter family or one of 9 exceptional supermanifolds. I singled out the homogeneous of these supermanifolds and described Lie superalgebras of vector fields on them.

Let $\mathbf L_k$ be the holomorphic line bundle of degree $k \in \mathbb Z$ on the projective line. Here, the tuples $(k_1 k_2 k_3 k_4)$ for which there does not exists homogeneous non-split supermanifolds $CP^{1|4}_{k_1 k_2 k_3 k_4}$ associated with the vector bundle $\mathbf L_{−k_1} \oplus \mathbf L _{−k_2} \oplus \mathbf L_{−k_3} \oplus \mathbf L_{−k_4}$ are classified. \\For many types of the remaining tuples, there are listed cocycles that determine homogeneous non-split supermanifolds. \\Proofs follow the lines indicated in the paper Bunegina V.A., Onishchik A.L., Homogeneous supermanifolds associated with the complex projective line.neous supermanifolds associated with the complex projective line. J. Math. Sci. V. 82 (1996)3503--3527.

All supermanifolds whose retract $T^{m|n}$ is determined by the trivial bundle of rank $n$ over the torus $T^m$ are $\overline 0$-homogeneous if and only if $T^{m|n}$ is homogeneous.

For any two root subsets associated with two Carter diagrams that have thesame $ADE$ type and the same size, we construct the transition matrix that mapsone subset to the other. The transition between these two subsets is carriedout in some canonical way affecting exactly one root, so that this root ismapped to the minimal element in some root subsystem. The constructedtransitions are involutions. It is shown that all root subsets associated withthe given Carter diagram are conjugate under the action of the Weyl group. Anumerical relationship is observed between enhanced Dynkin diagrams$\Delta(E_6)$, $\Delta(E_7)$ and $\Delta(E_8)$ (introduced by Dynkin-Minchenko)and Carter diagrams. This relationship echoes the $2-4-8$ assertions obtainedby Ringel, Rosenfeld and Baez in completely different contexts regarding theDynkin diagrams $E_6$, $E_7$, $E_8$.

Here are reproduced slightly edited notes of my lectures on theclassification of discrete groups generated by complex reflections of Hermitianaffine spaces delivered in October of 1980 at the University of Utrecht.