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Starting from the complete integrable lattice super-KdV equation, two super-mappings are obtained by performing a travelling-wave reduction. The first one is linear and the second is a four dimensional super-QRT mapping containing both Grassmann commuting and anti-commuting dependent variables. Adapting the classical "staircase" method to the Lax super-matrices of the lattice super-KdV equation, we compute the Lax super-matrices of the mapping and the two invariants; the first one is a pure nilpotent commuting quantity and the second one is given by an elliptic curve containing nilpotent commuting Grassmann coefficients as well. In the case of finitely generated Grassmann algebra with two generators, the super-QRT mapping becomes a four-dimensional ordinary discrete dynamical system that has two invariants but does not satisfy singularity confinement criterion. It is also observed that the dynamical degree of this system grows quadratically., Comment: 14 pages, no figures

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra G_m. A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G_2n, n = 2,3,4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{\bar\Psi},{\Psi}] = G_0[{\lambda\bar\Psi}, {\lambda\Psi}] + const., 0, Comment: 58 pages LaTeX (v2: mainly, minor updates and corrections to the reference section; v3: references [4], [17]-[21], [39], [46], [49]-[54], [61], [64], [139] added)

The deformation of the associative algebra of exterior forms is performed. This operation leads to a Yang–Baxter equation. Its relation with the braid group Bn−1 is analyzed. The correspondence of this deformation with the GLq(r,C) algebra is developed.

The nonstandard (infinitesimal) analysis developed by Robinson is applied to certain foundational aspects of superanalysis. The construction of nonstandard hull due to Luxemburg is used to study ‘‘limit phenomena’’ arising in a Grassmann algebra Λ(q) as the number of generators q tends to infinity. An outcome of such an approach is that under transition t→∞ a superspace of purely odd dimension (0,q) transforms into a superspace of dimension (∞,∞) and not (0,∞) as one would expect. It is suggested that this construction may have something to do with the problem of adequate mathematical description of collective bosonic effects arising within infinite fermionic dimensions.

Since the notion of angular momentum is defined in any dimension by using the exterior product in Rn, one would guess that central force problems in any dimension are completely integrable, as it is known for n=2 or 3. This is proved explicitly in this paper, by constructing n first integrals independent and involution: the energy and some combinations of the angular momentum components. It is shown that these problems are always reduced to a two‐dimensional plane, and the invariant manifolds are topologically, the Cartesian product of those of the reduced problem times n−2 circle factors. It is also proved that for n≳2 these problems are degenerate in the sense that their Hamiltonians do not satisfy one of the hypotheses of the Kolmogorov–Arnold–Moser theorem for persistence of invariant tori, when one considers small perturbations.

Previously elaborated principles [J. Math. Phys. 31, 1783 (1990)] for dimensional reduction of models of gauged supergroups over supercoset spaces are applied to the exceptional superalgebras. The superspace is parametrized by a pair of Grassmann coordinates.

Using the exact results for integrals of exponentials of polynomials of Grassmann variables and the Feynman construction of path integrals, an approximate method is presented to calculate the thermodynamic quantities of a fermionic system in equilibrium with a reservoir at temperature T and chemical potential μ. To exemplify, the quantum anharmonic fermionic oscillator is considered in a lattice with N points, where N=2, 3, and 4.

A semigroup of N=1 superconformal transformations is introduced and analyzed. Noninvertible ones can describe transitions from body to soul and form a proper ideal containing a set of nilpotent transformations. The projective superspace is also considered. Transformations twisting the parity of a tangent space are brought in. They can be a nonsuperconformal ‘‘square root’’ of noninvertible superconformal transformations and the analogs of the Poincare metric and conformal invariance are suggested for them.

Nonlinear superequations, for which the general solution can be expressed algebraically in terms of a finite number of particular solutions, are obtained. They are based on the orthosymplectic supergroup OSP(m,2n) and its action on a homogeneous superspace. Superposition formulas are discussed for the cases m=1, n arbitrary, and m=2, n=1. For OSP(2,2) the number of particular solutions needed to reconstruct the general solution depends on the dimension of the underlying Grassmann algebra, whereas for OSP(1,2n) it does not.

We show the existence of R matrices acting on the tensor product of a certain class of representations of the quantum toroidal algebra Uq(sl2,tor). In particular, the explicit expressions of R matrices acting on the tensor product of level 1 integrable highest weight representations of Uq(sl2∧) are obtained. Our approach is based on the work of Chari and Pressley.

A functional integral algorithm for the partition function and for the temperature Green’s function generating functional of spin 1/2 systems is developed. Starting from the representation of the spin operators in terms of fermion creation and destruction operators we write the above mentioned objects as functional integrals over an infinite dimensional Grassmann algebra. We show how to eliminate the integration over the Grassmann algebra in favor of an integration over function space. The use of the formalism is illustrated with two simple examples.

Definitions are given for the exterior product, infinitesimal extension of a cell, Levi‐Civita symbol, and generalized Kronecker delta in order to identify invariant integrals in spaces with both Fermi and Bose dimensions. Stokes’ and Green’s theorems for such spaces are constructed as a preliminary to defining a generalized action principle in supersymmetry spaces not necessarily equipped with a metric or a connection.

The relation between quaternionlike algebras and cross products of vectors is demonstrated. A classification of all cross products of vectors is given.

The prolongation structure method of Wahlquist and Estabrook is generalized to Grassmann algebra valued differential forms and used to determine a Backlund transformation for the equations of the anticommuting massive Thirring model.

A detailed construction is given of a supermanifold which is compact and has far from trivial topology. Higher dimensional examples are also described. The Grassmann algebra from which the supermanifolds are constructed may have finite or infinite dimension.