arXiv subjects

Sofiane Bouarroudj

Publications and source records attributed to Sofiane Bouarroudj.

At least 37 records · Page 2Linked to original sources

arXiv2022

Double and Lagrangian extensions for quasi-Frobenius Lie superalgebras

A Lie superalgebra is called quasi-Frobenius if it admits a closed anti-symmetric non-degenerate bilinear form. We study the notion of double extensions of quasi-Frobenius Lie superalgebra when the form is either orthosymplectic or periplectic. We show that every quasi-Frobenius Lie superalgebra that satisfies certain conditions can be obtained as a double extension of a smaller quasi-Frobenius Lie superalgebra. We classify all 4-dimensional quasi-Frobenius Lie superalgebras, and show that such Lie superalgebras must be solvable. We study the notion of $T^*$-extensions (or Lagrangian extensions) of Lie superalgebras, and show that they are classified by a certain cohomology space we introduce. Several examples are provided to illustrate our construction.

math.RT↗

arXiv2022

Analogs of Bol operators for $\mathfrak{pgl}(a+1\vert b)\subset \mathfrak{vect}(a\vert b)$

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.

math.RT↗

arXiv2021

Analogs of Bol operators on superstrings

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.

math.RT↗

arXiv2020

Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs - new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergence-free vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.

math.RT↗

arXiv2020

Double extensions of restricted Lie (super)algebras

A double extension ($\mathscr{D}$ extension) of a Lie (super)algebra $\mathfrak a$ with a non-degenerate invariant symmetric bilinear form $\mathscr{B}$, briefly: a NIS-(super)algebra, is an enlargement of $\mathfrak a$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the best known examples of double extensions of loops algebras. Let $\mathfrak a$ be a restricted Lie (super)algebra with a NIS $\mathscr{B}$. Suppose $\mathfrak a$ has a restricted derivation $\mathscr{D}$ such that $\mathscr{B}$ is $\mathscr{D}$-invariant. We show that the double extension of $\mathfrak a$ constructed by means of $\mathscr{B}$ and $\mathscr{D}$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $\mathscr{D}$-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $\mathscr{D}$-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.

math.RT↗

arXiv2019

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

We answered the 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 o(4) by means of the Poisson brackets? We presented a system which is not centrally symmetric, but has such 6 first integrals. We showed also that not every mechanical system with 3 degrees of freedom possesses such Lie algebra o(4).

physics.class-ph↗

arXiv2019

The roots of exceptional modular Lie superalgebras with Cartan matrix

For each of the exceptional Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots of and the corresponding Chevalley basis for one of the inequivalent Cartan matrices, the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots. Our main tools: Grozman's Mathematica-based code SuperLie, and Python.

math.RT↗

arXiv2018

Restricted simple Lie (super)algebras in characteristic $3$

We give explicit formulas proving restrictedness of the following Lie (super)algebras: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, and (under certain conditions) their simple subquotients over an algebraically closed field of characteristic 3, as well as one type of the deformed divergence-free Lie superalgebras with any number of indeterminates in any characteristic.

math.RT↗

arXiv2018

Classification of simple Lie superalgebras in characteristic $2$

All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures, so we classified all simple finite-dimensional Lie superalgebras modulo non-existing at the moment classification of simple finite-dimensional Lie algebras. This result concerns Lie superalgebras considered naively, as vector spaces. To obtain classification of simple Lie superalgebras in the category of supervarieties, one should list the non-isomorphic deforms (results of deformations) with odd parameter. This problem is open bar several examples described in arXiv~0807.3054. For Lie algebras, in addition to the known ---"classical" --- restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and of Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: $(2|4)$- and $(2|2)$-structures, one more analog --- a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras.

math.RT↗

arXiv2018

Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac--Moody (super)algebras are the most known examples.

math.RT↗

arXiv2018

Double extensions of Lie superalgebras in characteristic 2 with nondegenerate invariant supersymmetric bilinear form

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form will be called a NIS-Lie (super)algebra. The double extension of a NIS-Lie (super)algebra is the result of simultaneously adding to it a central element and an outer derivation so that the larger algebra has also a NIS. Affine loop algebras, Lie (super)algebras with symmetrizable Cartan matrix over any field, Manin triples, symplectic reflection (super)algebras are among the Lie (super)algebras suitable to be doubly extended. We consider double extensions of Lie superalgebras in characteristic 2, and concentrate on peculiarities of these notions related with the possibility for the bilinear form, the center, and the derivation to be odd. Two Lie superalgebras we discovered by this method are indigenous to the characteristic 2.

math.RT↗

arXiv2016

The classical dynamic symmetry for the $\mathrm{Sp}(1)$-Kepler problems

A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$ on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea due to A. Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of $\mathfrak {so}^*(4n)$ on the Poisson manifold $T^*\mathbb H_*^n/\mathrm{Sp}(1)$. (Here $\mathbb H_*^n:=\mathbb H^n\backslash \{0\}$ and the Hamiltonian action of $\mathrm{Sp}(1)$ on $T^*\mathbb H_*^n$ is induced from the natural right action of $\mathrm{Sp}(1)$ on $\mathbb H_*^n$. )

math-ph↗

arXiv2016

Invariant differential operators in positive characteristic

We consider an analog of the problem Veblen formulated in 1928 at the IMC: classify invariant differential operators between "natural objects" (spaces of either tensor fields, or jets, in modern terms) over a real manifold of any dimension. For unary operators, the problem was solved by Rudakov (no nonscalar operators except the exterior differential); for binary ones, by Grozman (there are no operators of orders higher than 3, operators of order 2 and 3 are, bar an exception in dimension 1, compositions of order 1 operators which, up to dualization and permutation of arguments, form 8 families). In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblen's problem in the 1-dimensional case over the ground field of positive characteristic. In addition to analogs of the Berezin integral (strangely overlooked so far) and binary operators constructed from them, we discovered two more (up to dualization) types of indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators.

math.RT↗

arXiv2015

The classical dynamic symmetry for the $\mathrm{U}(1)$-Kepler problems

For the Jordan algebra of hermitian matrices of order $n\ge 2$, we let $X$ be its submanifold consisting of rank-one semi-positive definite elements. The composition of the cotangent bundle map $π_X$: $T^*X\to X$ with the canonical map $X\to \mathbb{C}P^{n-1}$ (i.e., the map that sends a hermitian matrix to its column space), pulls back the Kähler form of the Fubini-Study metric on $\mathbb{C}P^{n-1}$ to a real closed differential two-form $ω_K$ on $T^*X$. Let $ω_X$ be the canonical symplectic form on $T^*X$ and $μ$ be a real number. A standard fact says that $ω_μ:=ω_X+2μ\,ω_K$ turns $T^*X$ into a symplectic manifold, hence a Poisson manifold with Poisson bracket $\{\, ,\,\}_μ$. In this article we exhibit a Poisson realization of the simple real Lie algebra $\mathfrak {su}(n, n)$ on the Poisson manifold $(T^*X, \{\, ,\,\}_μ)$, i.e., a Lie algebra homomorphism from $\mathfrak {su}(n, n)$ to $\left(C^\infty(T^*X, \mathbb R), \{\, ,\,\}_μ\right)$. Consequently one obtains the Laplace-Runge-Lenz vector for the classical $\mathrm{U}(1)$-Kepler problem with level $n$ and magnetic charge $μ$. Since the McIntosh-Cisneros-Zwanziger-Kepler problems (MICZ-Kepler Problems) are the $\mathrm{U}(1)$-Kepler problems with level $2$, the work presented here is a direct generalization of the work by A. Barut and G. Bornzin [ J. Math. Phys. $\bf 12$ (1971), 841-843] on the classical dynamic symmetry for the MICZ-Kepler problems.

math.DG↗

arXiv2015

Lie algebra deformations in characteristic 2

Of four types of Kaplansky algebras, type-2 and type-4 algebras have previously unobserved $\mathbb{Z}/2$-gradings: nonlinear in roots. A method assigning a simple Lie superalgebra to every $\mathbb{Z}/2$-graded simple Lie algebra in characteristic 2 is illustrated by seven new series. Type-2 algebras and one of the two type-4 algebras are demystified as nontrivial deforms (the results of deformations) of the alternate Hamiltonian algebras. The type-1 Kaplansky algebra is recognized as the derived of the nonalternate version of the Hamiltonian Lie algebra, the one that preserves a tensorial 2-form, not an exterior one. Deforms corresponding to nontrivial cohomology classes can be isomorphic to the initial algebra, e.g., we confirm Grishkov's implicit claim and explicitly describe the Jurman algebra as such a "semitrivial" deform of the derived of the alternate Hamiltonian Lie algebra. This paper helps to sharpen the formulation of a conjecture describing all simple finite-dimensional Lie algebras over any algebraically closed field of nonzero characteristic and supports a conjecture of Dzhumadildaev and Kostrikin stating that all simple finite-dimensional modular Lie algebras are either of "standard" type or deforms thereof. In characteristic 2, we give sufficient conditions for the known deformations to be semitrivial.

math.RT↗

arXiv2014

Riemannian Curl in Contact Geometry

We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing $1$-form is obtained from a Killing vector field by lowering indices. We show that the contact Riemannian curl vanishes if the metric is of constant curvature and the contact structure is defined by a Killing $1$-form. We also show that the contact Riemannian curl has a strong similarity with the Schwarzian derivative since it depends only on the projective equivalence class of the metric. For the Laplace-Beltrami operator on a contact manifold, the contact Riemannian curl is proportional to the subsymbol defined in arXiv:1205.6562. We also show that the contact Riemannian curl vanishes on the (co)tangent bundle over a Riemannian manifold. This implies that the corresponding subsymbol of the Laplace-Beltrami operator is identically zero.

math.DG↗

arXiv2013

Simple prolongs of the non-positive parts of graded Lie algebras with Cartan matrix in characteristic 2

Over an algebraically closed fields, an alternative to the method due to Kostrikin and Shafarevich was recently suggested. It produces all known simple finite dimensional Lie algebras in characteristic p>2. For p=2, we investigate one of the steps of this method, interpret several other simple Lie algebras, previously known only as sums of their components, as Lie algebras of vector fields. One new series of exceptional simple Lie algebras is discovered, together with its "hidden supersymmetries". In characteristic 2, certain simple Lie algebras are "desuperizations" of simple Lie superalgebras. Several simple Lie algebras we describe as results of generalized Cartan prolongation of the non-positive parts, relative a simplest (by declaring degree of just one pair of root vectors corresponding to opposite simple roots nonzero) grading by integers, of Lie algebras with Cartan matrix are "desuperizations" of characteristic 2 versions of complex simple exceptional vectorial Lie superalgebras. We list the Lie superalgebras (some of them new) obtained from the Lie algebras considered by declaring certain generators odd. One of the simple Lie algebras obtained is the prolong relative to a non-simplest grading, so the classification to be obtained might be more involved than we previously thought.

math.RT↗

arXiv2012

Minkowski superspaces and superstrings as almost real-complex supermanifolds

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the collection of subspaces of the tangent space, and in every subspace a complex structure is given. An almost complex structure on a real supermanifold can be given by an even or odd operator; it is complex (without "always") if the suitable superization of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we define the circumcised analog of the Nijenhuis tensor. We compute it for the Minkowski superspaces and superstrings. The space of values of the circumcised Nijenhuis tensor splits into (indecomposable, generally) components whose irreducible constituents are similar to those of Riemann or Penrose tensors. The Nijenhuis tensor vanishes identically only on superstrings of superdimension 1|1 and, besides, the superstring is endowed with a contact structure. We also prove that all real forms of complex Grassmann algebras are isomorphic although singled out by manifestly different anti-involutions.

math.DG↗