384 research outputs found
Arkady Onishchik: on his life and work on supersymmetry
No full textSelected stories about the life of A. L. Onishchik, and a review of his contribution to the classification of non-split supermanifolds, in particular, supercurves a.k.a. superstrings; his editorial and educational work. A brief overview of his and his students\u27 results in supersymmetry, and their impact on other researchers.
Several open problems growing out of Onishchik\u27s research are presented, some of them are related with odd parameters of deformations and non-holonomic structures of supermanifolds important in physical models, such as Minkowski superspaces and certain superstrings.24p
Non-split supermanifolds associated with the cotangent bundle
No full textHere, I study the problem of classification of non-split supermanifolds
having as retract the split supermanifold , where is the
sheaf of holomorphic forms on a given complex manifold of dimension .
I propose a general construction associating with any -closed -form
on a supermanifold with retract which is non-split
whenever the Dolbeault class of is non-zero. In particular, this gives
a non-empty family of non-split supermanifolds for any flag manifold . In the case where is an irreducible compact Hermitian
symmetric space, I get a complete classification of non-split supermanifolds
with retract . For each of these supermanifolds, the 0- and
1-cohomology with values in the tangent sheaf are calculated. As an example, I
study the -symmetric super-Grassmannians introduced by Yu. Manin.Comment: 79 page
Invariant bilinear differential operators
No full textI classified bilinear differential operators acting in the spaces of tensor
fields on any real or complex manifold and invariant with respect to the
diffeomorphisms in 1980. Here I give the details of the proof.Comment: 49 pages, LaTe
Two problems in the theory of differential equations
No full text1) The differential equation considered in terms of exterior differential forms, as É.Cartan did, singles out a differential ideal in the supercommutative superalgebra of differential forms, hence an affine supervariety. In view of this observation, it is evident that every differential equation has a supersymmetry (perhaps trivial). Superymmetries of which (systems of) classical differential equations are missed yet?
2) Why criteria of formal integrability of differential equations are never used in practice
Seminar of Supersymmetries : volume 1 (edited by D. Leites)
No full textSupermanifold theory is a relatively new branch of mathematics. Ideas of supersymmetry appeared to resolve several hitherto seemingly unsolvable problems of theoretical physics and quickly flourished into a rich blend of differential and algebraic geometers with own deep problems. In this book there are presented basics of linear and general algebra in superspaces, elements of algebraic and differential geometers on supermanifolds. The book is saturated by open questions of various degree of complexity and will be useful to researchers (theoretical physicists and mathematicians) as well as (research) students.</p
EXAMPLES OF SIMPLE VECTORIAL LIE ALGEBRAS IN CHARACTERISTIC 2
No full textThe classification of simple finite dimensional modular Lie algebras over algebraically closed fields of characteristic p > 3 (described by the generalized Kostrikin-Shafarevich conjecture) being completed due to Block, Wilson, Premet and Strade (with contributions from other researchers) the next major classification problems are those of simple finite dimensional modular Lie algebras over fields of characteristic 3 and 2. For the latter, the Kochetkov-Leites conjecture involved classification of Lie superalgebras and their inhomogeneous with respect to parity subalgebras, called Volichenko algebras. In characteristic 2, we consider the result of application of the functor forgetting the superstructure to the simple serial vectorial Lie algebras known to us and their Volichenko subalgebras.</p
Inverses of Cartan matrices of Lie algebras and Lie superalgebras
No full textThe inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras, except for the 5 Lie superalgebras whose Cartan matrices have 0 on the main diagonal: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra may differ (in any characteristic).
We interpret most of the results of Wei Yangjiang and Zou Yi Ming, Inverses of Cartan matrices of Lie algebras and Lie superalgebras, Linear Alg. Appl., 521 (2017) 283--298 as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin\u27s paper in 1952, see also Table 2 in Springer\u27s book by Onishchik and Vinberg (1990).We reproduce definition of root spaces from arXiv:0710.5149 and arXiv:0906.1860. Version 2 contains inadvertently forgotten subsection 8.3 and accordingly edited Introduction and Abstrac
On odd parameters in geometry
No full text1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.~Bernstein and I, and independently M.~Duflo, observed that certain divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that other simple Lie superalgebras have no such deformations (unpublished). Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and (very sketchily) relations with simple Lie (super)algebras over fields of positive characteristic.
2) Any supermanifold which is a ringed space of the form (a manifold , the sheaf of sections of the exterior algebra of a vector bundle over ) is called split. Gawȩdzki (1977) and Batchelor (1979) proved that every smooth supermanifolds is split. In 1982, P. Green and Palamodov showed that a~complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a split supermanifold. So far, researchers considered, mostly, even obstructions to splitness. This lead them to the conclusion that any supermanifolds of superdimension is split. I\u27ll show that there are non-split supermanifolds of superdimension ; for example, certain -dimensional superstrings, the obstructions to their splitness correspond to odd parameters.44 pages; the strange words in Theorem 5.1 are striken out, a references updated; otherwise coincides with the published versio
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