arXiv subjects

Dimitry Leites

Publications and source records attributed to Dimitry Leites.

At least 55 records · Page 3Linked to original sources

arXiv2009

Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix

Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical, most probably inessential, hypotheses. If the Cartan matrix is invertible, the corresponding Lie superalgebra is simple otherwise the quotient of the derived Lie superalgebra modulo center is simple (if its rank is greater than 1). Eleven new exceptional simple modular Lie superalgebras are discovered. Several features of classic notions, or notions themselves, are clarified or introduced, e.g., Cartan matrix, several versions of restrictedness in characteristic 2, Dynkin diagram, Chevalley generators, and even the notion of Lie superalgebra if the characteristic is equal to 2. Interesting phenomena in characteristic 2: (1) all simple Lie superalgebras with Cartan matrix are obtained from simple Lie algebras with Cartan matrix by declaring several (any) of its Chevalley generators odd; (2) there exist simple Lie superalgebras whose even parts are solvable. The Lie superalgebras of fixed points of automorphisms corresponding to the symmetries of Dynkin diagrams are also listed and their simple subquotients described.

math.RT↗

arXiv2009

The classification of almost affine (hyperbolic) Lie superalgebras

We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine but the Lie (super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine, and all of its Cartan matrices are almost affine. We list all almost affine Lie superalgebras over complex numbers correcting two earlier claims of classification and make available the list of almost affine Lie algebras obtained by Li Wang Lai.

math.RA↗

arXiv2009

The Poincare series of the hyperbolic Coxeter groups with finite volume of fundamental domains

The discrete group generated by reflections of the sphere, or Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series (a.k.a. growth function) is the generating function of the cardinalities of sets of elements of equal length. Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave a recurrence formula. The explicit expression of the Poincaré series was known for the spherical and Euclidean Coxeter groups, and 3-generated Coxeter groups, and (with mistakes) Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, and denominators for each (quasi-)Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the Poincaré series. The non-real poles behave as in Eneström's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements.

math.RT↗

arXiv2007

Towards classification of simple finite dimensional modular Lie superalgebras

A way to construct (conjecturally all) simple finite dimensional modular Lie (super)algebras over algebraically closed fields of characteristic not 2 is offered. In characteristic 2, the method is supposed to give only simple Lie (super)algebras graded by integers and only some of the non-graded ones). The conjecture is backed up with the latest results computationally most difficult of which are obtained with the help of Grozman's software package SuperLie.

math.RT↗

arXiv2007

New simple modular Lie superalgebras as generalized prolongs

Over algebraically closed fields of characteristic p>2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified.

math.RT↗

arXiv2006

Cartan matrices and presentations of Cunha and Elduque superalgebras

All inequivalent Cartan matrices (in other words, inequivalent systems of simple roots) of the ten simple exceptional finite dimensional Lie superalgebras in characteristic 3, recently identified by Cunha and Elduque as constituents of Elduque's superization of the Freudenthal Magic Square, are listed together with defining relations between analogs of their Chevalley generators.

math.RT↗

arXiv2006

Cartan matrices and presentations of the exceptional simple Elduque Lie superalgebra

Recently Alberto Elduque listed all simple and graded modulo 2 finite dimensional Lie algebras and superalgebras whose odd component is the spinor representation of the orthogonal Lie algebra equal to the even component, and discovered one exceptional such Lie superalgebra in characteristic 5. For this Lie superalgebra all inequivalent Cartan matrices (in other words, inequivalent systems of simple roots) are listed together with defining relations between analogs of its Chevalley generators.

math.RT↗

arXiv2006

Simple Lie superalgebras and nonintegrable distributions in characteristic p

Recently, Grozman and Leites returned to the original Cartan's description of Lie algebras to interpret the Melikyan algebras (for p<7) and several other little-known simple Lie algebras over algebraically closed fields for p=3 as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4) and Sp(10). The description was performed in terms of Cartan-Tanaka-Shchepochkina prolongs using Shchepochkina's algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite dimensional Lie superalgebras over C; for p=3, we obtain a series of new simple Lie superalgebras and an exceptional one.

math.RT↗

arXiv2005

How to quantize the antibracket

The uniqueness of (the class of) deformation of Poisson Lie algebra has long been a completely accepted folklore. Actually, it is wrong as stated, because its validity depends on the class of functions that generate Poisson Lie algebra, Po(2n): it is true for polynomials but false for Laurent polynomials. We show that unlike the Lie superalgebra Po(2n|m), its quotient modulo center, the Lie superalgebra H(2n|m) of Hamiltonian vector fields with polynomial coefficients, has exceptional extra deformations for (2n|m)=(2|2) and only for this superdimension. We relate this result to the complete description of deformations of the antibracket (also called the Schouten or Buttin bracket). The representation of the deform (the result of quantization) of the Poisson algebra in the Fock space coincides with the simplest space on which the Lie algebra of commutation relations acts. This coincidence is not necessary for Lie superalgebras

math-ph↗

arXiv2005

Defining relations associated with the principal sl(2)-subalgebras of simple Lie algebras

The notion of defining relations is well-defined for any nilpotent Lie algebra. Therefore a conventional way to present a simple Lie algebra G is by splitting it into the direct sum of a commutative Cartan subalgebra and two maximal nilpotent subalgebras (positive and negative) and together the generators of both these nilpotent subalgebras together generate G. Though there are many relations between these generators, they are neat (Serre relations). It is possible to determine the relations for generators of different type, e.g, with the principal embeddings of sl(2) into G one can associate only TWO elements that generate G. We explicitly describe the corresponding presentations of simple Lie algebras, for all finite dimensional and certain infinite dimensional ones; namely, for the Lie algebra "of matrices of a complex size" realized as a subalgebra of the Lie algebra of differential operators in 1 indeterminate. The relations obtained are rather simple. Our results might be of interest in applications to integrable systems (like vector-valued Liouville (or Leznov-Saveliev, or 2-dimensional Toda) equations and KdV-type equations). They also indicate how to q-quantize the Lie algebra of matrices of complex size.

math-ph↗

arXiv2005

Defining relations for classical Lie algebras of polynomial vector fields

We explicitly describe the defining relations for simple Lie algebra of vector fields with polynomial coefficients and its subalgebras of divergence free, hamiltonian and contact vector fields, and for the Poisson algebra (realized on polynomials). We consider generators of these Lie algebras corresponding to the systems of simple roots associated with the standard grading of these algebras. (These systems of simple roots are distinguished in the sense of Penkov and Serganova.

math.RT↗

arXiv2005

The analogs of the Riemann tensor for exceptional structures on supermanifolds

H. Hertz called any manifold M with a given nonintegrable distribution {\it nonholonomic}. Vershik and Gershkovich stated and R. Montgomery proved that the space of germs of any nonholonomic distribution on M with an open and dense orbit of the diffeomorphism group is either (1) of codimension one or (2) an Engel distribution. No analog of this statement for supermanifolds is formulated yet, we only have some examples: our list (an analog of E.Cartan's classification) of simple Lie superalgebras of vector fields with polynomial coefficients and a particular (Weisfeiler) grading contains 16 series similar to contact ones and 11 exceptional algebras preserving nonholonomic structures. Here we compute the cohomology corresponding to the analog of the Riemann tensor for the SUPERmanifolds corresponding to the 15 exceptional simple vectorial Lie superalgebras, 11 of which are nonholonomic. The cohomology for analogs of the Riemann tensor for the manifolds with an exceptional Engel manifolds are computed in math.RT/0202213.

math.RT↗

arXiv2005

Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced.

math.RT↗

arXiv2005

Lie superalgebra structures in H*(g; g)

On a given manifold M, the Nijenhuis bracket makes the superspace of vector-valued differential forms into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in the superspaces of cochains and cohomology with coefficients in the adjoint module for any Lie superalgebra. We use a Mathematica--based package SuperLie (already proven useful in various problems) to explicitly describe these Lie superalgebras for some simple finite dimensional Lie superalgebras and their ``relatives'' (the nontrivial central extensions or derivation algebras of the considered simple ones).

math.RT↗

arXiv2004

Lie superalgebra structures in cohomology spaces of Lie algebras with coefficients in the adjoint representation

The space of Lie algebra cohomology is usually described by the dimensions of components of certain degree even for the adjoint module as coefficients when the spaces of cochains and cohomology can be endowed with a Lie superalgebra structure. Such a description is rather imprecise: these dimensions may coincide for cohomology spaces of distinct algebras. We explicitely describe the Lie superalgebras on the space of cohomology of the maximal nilpotent subalgebra of any simple finite dimensional Lie algebra. We briefly review related results by Grozman, Penkov and Serganova, Poletaeva, and Tolpygo. We cite a powerful Premet's theorem complementary to the Borel-Weil-Bott theorem. We mention relations with the Nijenhuis bracket and nonholonomic systems.

math.KT↗

arXiv2002

Invariant operators on supermanifolds and Standard Models

Here we continue to list the differential operators invariant with respect to the 15 exceptional simple Lie superalgebras of polynomial vector fields. A part of the list (for operators acting on tensors with finite dimensional fibers) was earlier obtained in 2 of the 15 cases by Kochetkov and in one more instance by Kac and Rudakov. Broadhurst and Kac conjectured that some of these structures pertain to the Standard Models of elementary particles and the Grand Unified Theories. So, GUT, if any exists, will be formulated in terms of operators we found, or their r-nary analogs to be found. Calculations are performed with the aid of Grozman's Mathematica-based SuperLie package. When degeneracy conditions are violated (absence of singular vectors) the corresponding module of tensor fields is irreducible. We consider generalized tensors corresponding to fibers of countable infinite dimension, not only finite dimensional ones. We also verified some of the earlier findings.

math.RT↗

arXiv2002

The Riemann tensor for nonholonomic manifolds

For every nonholonomic manifold, i.e., manifold with nonintegrable distribution, the analog of the Riemann tensor is introduced. It is calculated here for the contact and Engel structures: for the contact structure it vanishes (another proof of Darboux's canonical form); for the Engel distribution the target space of the tensor is of dimension 2. In particular, the Lie algebra preserving the Engel distribution is described. The tensors introduced are interpreted as modifications of the Spencer cohomology and, as such, provide with a new way to solve partial differential equations. Goldschmidt's criteria for formal integrability (vanishing of certain Spencer cohomology) are only applicable to ``one half'' of all differential equations, the ones whose symmetries are induced by point transformations. Lie's theorem says that the ``other half'' consists of differential equations whose symmetries are induced by contact transformations. Therefore, we can now extend Goldschmidt's criteria for formal integrability to all differential equations.

math.RT↗

arXiv2002

On unconventional integrations and cross ratio on supermanifolds

The conventional integration theory on supermanifolds had been constructed so as to possess (an analog of) Stokes' formula. In it, the exterior differential d is vital and the integrand is a section of a fiber bundle of finite rank. Other, not so popular, but, nevertheless, known integrations are analogs of Berezin integral associated with infinite dimensional fibers. Here I offer other unconventional integrations that appear thanks to existence of several versions of traces and determinants and do not allow Stokes formula. Such unconventional integrations have no counterpart on manifolds except in characteristic p. Another type of invariants considered are analogs of the cross ratio for ``classical superspaces''. As a digression, homological fields corresponding to simple Lie algebras and superalgebras are described.

math.RT↗