In the Belinski-Khalatnikov-Lifshitz cosmological models with an oscillatory approximation to singularity, the tracks of cosmological billiards (whatever they are) are realized in the Weyl chambers of the hyperbolic Lie algebras. The latter were classified by Li Wang Lai; their super analogs — the almost affine Lie superalgebras — were classified in arXiv:0906.1860. Here, we observe that some of the 142 hyperbolic Lie algebras H can be ''superized'' to almost affine Lie superalgebras S(H): for each of these H, it is possible to divide a row of its Cartan matrix by 2, simultaneously changing the parity of the corresponding Chevalley generators and relations between them. For the H with a symmetrizable Cartan matrix, we list all 97 such pairs (H⟷S(H)). Several (18 of the total 66) thus ''superizable'' algebras H have multiple such ''superizations''. The tracks of cosmological billiards corresponding to both terms of these pairs (H⟷S(H)) coincide since H and S(H) have one and the same Weyl chamber. We also classify ''superizations'' of hyperbolic Lie algebras with non-symmetrizable Cartan matrix, and pairs , where g is a simple finite-dimensional Lie superalgebra without Cartan matrix and simple component .

The two classifications of bilinear differential operators are often mixed in the literature:(A) Between spaces of modular forms. These forms are transformed under SL(2;Z) as weighted densities of (usually integer) weight and satisfy certain growth conditions so their spaces are finite-dimensional. The SL(2;Z)-invariant bilinear differential brackets between spaces of modular forms on the 1-dimensional manifold were discovered by Aronhold and Gordan, rediscovered and classified by Rankin and Cohen, and by Janson and Peetre.(B) Between spaces of weighted densities. Here, for any complex weights, we classify bilinear differential operators between (infinite-dimensional) spaces of weighted densities, i.e., the pgl(2)≃sl(2)-invariant operators on the line. This is a new result.

Any supermanifold diffeomorphic to one whose structure sheaf is the sheaf of sections of the exterior algebra of a vector bundle E over the underlying manifold M is called split. Gawȩdzki (1977) and Batchelor (1979) were the first to prove that any smooth supermanifold is split. In 1981, P. Green, and Palamodov, found examples of non-split analytic supermanifolds and described obstructions to splitness that were further studied by Manin (resp., Onishchik with his students) following Palamodov's (resp., Green's) approach. Following Palamodov, Donagi and Witten demonstrated that some of the moduli supervarieties of superstring theories are non-split. Except for arXiv:2210.17096, the odd parameters of supervarieties of obstructions to splitness were never considered. Here, using Palamodov's approach, we classify and describe the even (degree-2) and the odd (degree-1) obstructions to splitness of (1|2)-dimensional superstrings. In particular, we correct calculations of degree-2 obstructions due to Bunegina and Onishchik and confirm Manin's answer, but correct his description of the group Aut(E).

We describe supertraces on “queerifications” (see arXiv:2203.06917) of the algebras of matrices of “complex size”, algebras of observables of Calogero–Moser model, Vasiliev higher spin algebras, and (super)algebras of pseudo-differential operators. In the latter case, the supertraces establish complete integrability of the analogs of Euler equations to be written (this is one of several open problems and conjectures offered).

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank <9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8—most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold),—we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras—the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

(A) The conventional integration theory on supermanifolds had been constructed in order to have (an analog of) the Stokes formula in which a sub-supermanifold is of codimension 1 = (1|0). I review other integrations and formulate related open problems:

(1) On the 1|1-dimensional superstring associated with the trivial bundle, in presence of a contact structure there is a special integration useful in describing super versions of elliptic functions. It is needed to construct a~particular spinor representation of the Neveu-Schwarz superalgebra.

(2) Versions of the Stokes formula with "over-supermanifold" of codimension (0|-1) due to Shander and Palamodov should be developed further.

(3) Apply Shander's integration with odd parameters over chains to inverse problems.

(4) Establish existence of conjectural integrations (apparently, not leading to any analog of the Stokes formula) related to various (super)traces on various Lie superalgebras and the corresponding (super)determinants.

(B) I offer analogs of the cross ratio for "classical superspaces'', including infinite-dimensional versions. Open problem: apply these invariants to the matrix-valued Riccati equations.

1) 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 M, the sheaf of sections of the exterior algebra of a vector bundle over M) 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 m|1 is split. I'll show that there are non-split supermanifolds of superdimension m|1; for example, certain 1|1-dimensional superstrings, the obstructions to their splitness correspond to odd parameters.

Bol operators (Bols for short) are differential operators invariant under the projective action of pgl(2)≃sl(2)𝔭𝔤𝔩(2)≃𝔰𝔩(2) between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: pgl(a+1|b)𝔭𝔤𝔩(a+1|b)-invariant differential operators between spaces of tensor fields on (a|b)(a|b)-dimensional supermanifolds with irreducible, as gl(a|b)𝔤𝔩(a|b)-modules, fibers of arbitrary, even infinite, dimension for certain “key” values of aa and bb — the ones for which the solution is describable. We discovered many new operators for (a|b)=(2|0),(0|3)(a|b)=(2|0),(0|3) and for the case of 1|11|1-dimensional general superstring, which looks like a most natural superization of Bol’s result, additional to the cases of super analogs of Bols between spaces of weighted densities on the 1|n1|n-dimensional superstrings with a contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension >1>1, there are (a+b−1)(a+b−1)-parameter families of Bols, whereas there are no non-scalar nonzero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here.

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the 1 vertical bar n-dimensional supermanifold (superstring) M, we classify analogs of Bol operators invariant under the simple maximal subalgebra h of the same rank as its simple ambient superalgebra g of vector fields on M and containing all elements of negative degree of g in a Z-grading. We also consider the Lie superalgebras of vector fields g preserving a contact structure on the superstring M. We have discovered many new operators.