arXiv subjects
Publications and source records attributed to Dimitry Leites.
The Steenrod algebra can not be realised as an enveloping of any Lie superalgebra. We list several problems that suggest a need to modify the definition of the enveloping algebra, for example, to get rid of certain strange deformations which we qualify as an artefact of the inadequate definition of the enveloping algebra in positive characteristic. P. Deligne appended our paper with his comments, hints and open problems.
Any supermanifold diffeomorphic to one whose structure sheaf is the sheaf of sections of a~vector bundle over the underlying manifold is called split. Gaw\c{e}dzki (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. None of the above-mentioned authors considered odd parameters of supervarieties of obstructions to non-splitness. 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.
We overview classifications of simple infinite-dimensional complex $\mathbb{Z}$-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose elements are vector fields with polynomial, or formal power series, or divided power coefficients. A given vectorial Lie (super)algebra with a (Weisfeiler) filtration corresponding to a maximal subalgebra of finite codimension is called W-filtered; the associated graded algebra is called W-graded. Here, we correct several published results: (1) prove our old claim "the superization of \'E. Cartan's problem (classify primitive Lie algebras) is wild", (2) solve a tame problem: classify simple W-graded and W-filtered vectorial Lie superalgebras, (3) describe the supervariety of deformation parameters for the serial W-graded simple vectorial superalgebras, (4) conjecture that the exceptional simple vectorial superalgebras are rigid. We conjecture usefulness of our method in classification of simple infinite-dimensional vectorial Lie (super)algebras over fields of positive characteristic.
The equations of open 2-dimensional Toda lattice (TL) correspond to Leznov-Saveliev equations (LSE) interpreted as zero-curvature Yang-Mills equations on the variety of $O(3)$-orbits on the Minkowski space when the gauge algebra is the image of $\mathfrak{sl}(2)$ under a principal embedding into a simple finite-dimensional Lie algebra $\mathfrak{g}(A)$ with Cartan matrix $A$. The known integrable super versions of TL equations correspond to matrices $A$ of two different types. I interpret the super LSE of one type 1 as zero-curvature equations for the \textit{reduced} connection on the non-integrable distribution on the supervariety of $OSp(1|2)$-orbits on the $N=1$-extended Minkowski superspace; the Leznov-Saveliev method of solution is applicable only to $\mathfrak{g}(A)$ finite-dimensional and admitting a superprincipal embedding $\mathfrak{osp}(1|2)\to\mathfrak{g}(A)$. The simplest LSE1 is the super Liouville equation; it can be also interpreted in terms of the superstring action. Olshanetsky introduced LSE2 -- another type of equations of super TL. Olshanetsky's equations, as well as LSE1 with infinite-dimensional $\mathfrak{g}(A)$, can be solved by the Inverse Scattering Method. To interpret these equations remains an open problem, except for the super Liouville equation -- the only case where these two types of LSE coincide. I also review related less known and less popular mathematical constructions involved.
Under usual locality assumptions, we classify all non-integrable distributions with simple infinite-dimensional Lie superalgebra of symmetries over $\mathbb{C}$: we single out 15 series (containing 2 analogs of contact series and one family of deformations of their divergence-free subalgebras), and 7 exceptional Lie superalgebras. Over algebraically closed fields~$\mathbb{K}$ of characteristic $p>0$, we classify the W-gradings (corresponding to a maximal subalgebra of finite codimension) of the known simple vectorial Lie (super)algebras with unconstrained shearing vector of heights of the indeterminates, distinguish W-gradings of (super)algebras preserving non-integrable distributions. For $p>3$, we get analogs of the result over $\mathbb{C}$. For $p=3$, of all possible W-gradings (12 of Skryabin algebras, 3 of superized Melikyan algebras, and 4 of Bouarroudj superalgebras) most are new, together with the corresponding distributions. For $p=2$, we also get several new examples of distributions and their Lie (super)algebras of symmetries.
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.
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. Gaw\c{e}dzki (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.
Over $\mathbb{C}$, Montgomery superized Herstein's construction of simple Lie algebras from finite-dimensional associative algebras, found obstructions to the procedure and applied it to $\mathbb{Z}/2$-graded associative algebra of differential operators with polynomial coefficients. Since the 1990s, Vasiliev and Konstein with their co-authors constructed (via the Herstein--Montgomery method, having rediscovered it) simple Lie (super)algebras from the associative (super)algebra such as Vasiliev's higher spin algebras (a.k.a. algebras of observables of the rational Calogero model) and algebras of symplectic reflections. The "queerification" is another method for cooking a~simple Lie superalgebra from the simple associative (super)algebra. The above examples of associative (super)algebras, and Lie (super)algebras of "matrices of complex size" can be "queerified" by adding new elements resembling Faddeev--Popov ghosts. Conjectures: 1) a "queerified" Hamiltonian describes a version of the Calogero model with $1\vert 1$-dimensional time; 2) metabelean algebras and inhomogeneous subalgebras of Lie superalgebras naturally widen supersymmetries in future theories; 3) only graded-commutative algebras can imitate algebras of functions in a reasonably rich non-commutative Geometry.
The Amitsur--Levitzki identity for matrices was generalized in several directions: by Kostant for simple finite-dimensional Lie algebras, by Kirillov (later joined by Kontsevich, Molev, Ovsienko, and Udalova) for simple vectorial Lie algebras with polynomial coefficients, and by Gie, Pinczon, and Ushirobira for the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|n)$. Dzhumadildaev switched the focus of attention in these results by considering the algebra formed by antisymmetrizors and discovered a hidden supersymmetry of commutators. We overview these results and their possible generalizations (open problems).
Selected 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' results in supersymmetry, and their impact on other researchers. Several open problems growing out of Onishchik's 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.
Bol operators (Bols for short) are differential operators invariant under the projective action of $\mathfrak{pgl}(2)\simeq\mathfrak{sl}(2)$ between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: $\mathfrak{pgl}(a+1\vert b)$-invariant differential operators between spaces of tensor fields on $(a\vert b)$-dimensional supermanifolds with irreducible, as $\mathfrak{gl}(a\vert b)$-modules, fibers of arbitrary, even infinite, dimension for certain ``key" values of $a$ and $b$ -- the ones for which the solution is describable. We discovered many new operators for $(a|b)=(2|0), (0|3)$ and for the case of $1\vert 1$-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 $1\vert n$-dimensional superstrings with a~contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension $>1$, there are $(a+b-1)$-parameter families of Bols, whereas there are no non-scalar non-zero 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 $1|n$-dimensional supermanifold (superstring) $\mathcal{M}$, we classify analogs of Bol operators invariant under the simple maximal subalgebra $\mathfrak{h}$ of the same rank as its simple ambient superalgebra $\mathfrak{g}$ of vector fields on $\mathcal{M}$ and containing all elements of negative degree of $\mathfrak{g}$ in a $\mathbb{Z}$-grading. We also consider the Lie superalgebras of vector fields $\mathfrak{g}$ preserving a contact structure on the superstring $\mathcal{M}$. We have discovered many new operators.
As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2. We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or $0|1$, or $1|1$; it is equal to $1|1$ if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic distinct from 2, see arXiv:1806.05505). We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2. We also recall examples of multiple NISes on simple Lie algebras over non-closed fields.
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 ($H\longleftrightarrow S(H)$). 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 ($H\longleftrightarrow S(H)$) 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 $\mathfrak{g}\longleftrightarrow \mathfrak{g}_{\bar 0}$, where $\mathfrak{g}$ is a simple finite-dimensional Lie superalgebra without Cartan matrix and simple component $\mathfrak{g}_{\bar 0}$.
For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square. These homologies are~key ingredients in the Duflo--Serganova approach to the representation theory. There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. In positive characteristic, we found out one more reason to consider the space of roots over reals, unlike the space of weights, which should be considered over the ground field. We proved that the rank of the homological element (decisive in calculating the defect of a given Lie superalgebra) should be considered in the adjoint module, not the irreducible module of least dimension (although the latter is sometimes possible to consider, e.g., for $p=0$). We also computed the above homology for the only case of simple Lie superalgebras with symmetric root system not considered so far over the field of complex numbers, and its modular versions: $\mathfrak{psl}(a|a+pk)$ for $a$ and $k$ small, and $p=2, 3, 5$.
The definition of Kaehler manifold is superized. In the super setting, it admits a continuous parameter, unlike their analogs on manifolds. This parameter runs the same singular supervariety of parameters that parameterize deformations of the Schouten bracket (a.k.a. Buttin bracket, a.k.a. anti-bracket) considered as deformations of the Lie superalgebra structure given by the bracket. The same idea yields definitions of several versions of hyper-Kaeahler supermanifolds depending on parameters that also run over a singular supervariety. Moreover, the same idea is potentially applicable to the Kaehler and hyper-Kaehler manifolds (or supermanifolds corresponding to the even tensors that define them); in these cases infinite-dimensional (super)manifolds should enter the picture. Strangely enough, "how to embody this idea for the case of only even tensors involved?" is an open problem. The actions of Lie algebras on the space of differential forms on symplectic, and hyper-Kaehler manifold (known already to A.Weil, and Verbitsky, respectively) are extended to actions of Lie superalgebras on the same spaces with values in a line bundle with a maximally non-integrable connections, see Leites D., Shchepochkina I., The Howe duality and Lie superalgebras. In: S.~Duplij and J.~Wess (eds.) "Noncommutative Structures in Mathematics and Physics", Proc. NATO Advanced Research Workshop, Kiev, 2000. Kluwer, 2001, 93--112; arXiv:math.RT/0202181.
The Amitsur-Levitzki identity for matrices was generalized in several directions: by Kostant for simple finite-dimensional Lie algebras, by Kirillov (later joined by Kontsevich, Molev, Ovsienko, and Udalova) for simple vectorial Lie algebras with polynomial coefficients, and by Gie, Pinczon, and Ushirobira for the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|n)$. Dzhumadildaev switched the focus of attention in these results by considering the algebra formed by antisymmetrizors and discovered a hidden supersymmetry of commutators. We overview these results and their possible generalizations (open problems).
The 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's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990).