Journal of Nonlinear Mathematical Physics

A boson-fermion correspondence allows an analytic definition of functional super derivative, in particular, and a bosonic functional calculus, in general, on Bargmann Gelfand triples for the second super quantization. A Feynman integral for the super transformation matrix elements in terms of bosonic anti-normal Berezin-Sudarshan symbols is rigorously constructed.

We show that a certain stochastic perturbation of the flow of perfect incompressible fluid under some special external force on the flat n-dimensional torus yields a solution of Navier–Stokes equation without external force in the tangent space at unit of volume preserving diffeomorphism group. If that external force is absent, the equation turns into the one of Reynolds type. For the flow of diffuse matter this construction yields the Burgers equation.

A series of bilinear identities on the Schur symmetric functions is obtained with the use of Plücker relations.

Let be the n-dimensional pseudo-Euclidean space of index p and M(n, p) the group of all transformations of generated by pseudo-orthogonal transformations and parallel translations. We describe the system of generators of the differential field of all M(n, p)-invariant differential rational functions of a map of an open subset . Using this result, we prove analogues of the Bonnet theorem for immersions of an n-dimensional C^∞-manifold J in . These analogues are given in terms of the pseudo-Riemannian metric, the volume form, and the connection on J induced by the immersion of J in .

I discuss the connection of the three different questions: The existence of the Gibbs steady state distributions for the stochastic differential equations, the notion and the existence of the conservation laws for such equations, and the convergence of the smooth random perturbations of dynamical systems to stochastic differential equations in the Ito sense. I show that in all cases one needs to include some additional term in the standard form of stochastic equation. I call such approach to describing the influence of the noise on the dynamical systems the "stochastic deformation" to distinguish it from the conventional "stochastic perturbation". I also discuss some consequences of this approach, in particular, a connection between the noise intensity and the temperature. This connection is known in physics (for the case of linear system of differential equations) as "fluctuation-dissipation theorem" (L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 9, Statistical Physics. Part 2). In conclusion, I present an interesting physical example of the dynamics of magnetic dipole in a random magnetic field.

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac–Moody algebras.

We say that an indecomposable Cartan matrix A with entries in the ground field is almost affine if the Lie (super)algebra determined by it is not finite dimensional or affine (Kac–Moody) but the Lie sub(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 (Kac–Moody), and all of its Cartan matrices are almost affine.

We list all almost affine Lie superalgebras over complex numbers with indecomposable Cartan matrix correcting two earlier claims of classification.

For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators.

The discrete group generated by reflections of the sphere, or the 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 is the generating function of the cardinalities of sets of elements of equal length. Around 1966, 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 an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. 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.

Over algebraically closed fields of characteristic 2, the analogs of the orthogonal, symplectic, Hamiltonian, Poisson, and contact Lie superalgebras are described. The number of non-isomorphic types, and several properties of these algebras are unexpected, for example, interpretation in terms of exterior differential forms preserved is not applicable to one of these types. The divided powers of differential forms and related (co)homology are introduced.

Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial) are, if the depth of their grading is greater than 1, results of generalized Cartan–Tanaka–Shchepochkina (CTS) prolongs.

Here we are looking for new examples of simple finite dimensional modular Lie (super)algebras in characteristic 2 obtained as Cartan prolongs. We consider pairs (an (ortho-)orthogonal Lie (super)algebra or its derived algebra, its irreducible module) and compute the Cartan prolongs of such pairs. The derived algebras of these prolongs are simple Lie (super)algebras.

We point out several amazing phenomena in characteristic 2: a supersymmetry of representations of certain Lie algebras, latent or hidden over complex numbers, becomes manifest; the adjoint representation of some simple Lie superalgebras is not irreducible.

The 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.

Here the theory of finite-dimensional supermanifolds is generalized in two directions.

First, we introduce infinite-dimensional supermanifolds "locally isomorphic" to arbitrary Banach (or, more generally, locally convex) superspaces. This is achieved by considering supermanifolds as functors (equipped with some additional structure) from the category of finite-dimensional Grassmann superalgebras into the category of the corresponding smooth manifolds (Banach or locally convex).

Selected examples: flag supermanifolds of Banach superspaces as well as unitary supergroups of Hilbert superspaces.

Second, we define "generalized" supermanifolds, "locally isomorphic" to -graded Banach (or, more generally, locally convex) spaces. These generalized supermanifolds are referred to in what follows (super stringsol;)^k-manifolds, or colored manifolds). The corresponding superfields (i.e., morphisms into "coordinate ring") describing, hopefully, particles with more general statistics than Bose + Fermi, turn out to have, in general, an infinite number of components.