Conserved currents for unconventional supersymmetric couplings of self-dual gauge fields

Self-dual gauge potentials admit supersymmetric couplings to higher-spin fields satisfying interacting forms of the first order Dirac--Fierz equation. The interactions are governed by conserved currents determined by supersymmetry. These super-self-dual Yang-Mills systems provide on-shell supermultiplets of arbitrarily extended super-Poincar\'e algebras; classical consistency not setting any limit on the extension N. We explicitly display equations of motion up to the $N=6$ extension. The stress tensor, which vanishes for the $N\le 3$ self-duality equations, not only gets resurrected when $N=4$, but is then a member of a conserved multiplet of gauge-invariant tensors.Comment: 6 pages, latex fil

R-matrices in Rime

We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by a weaker condition which we call "rime". Rime solutions include the standard Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime Ansatz which are maximally different from the standard one we call "strict rime". A strict rime non-unitary solution is parameterized by a projective vector. We show that this solution transforms to the Cremmer-Gervais R-matrix by a change of basis with a matrix containing symmetric functions in the components of the parameterizing vector. A strict unitary solution (the rime Ansatz is well adapted for taking a unitary limit) is shown to be equivalent to a quantization of a classical "boundary" r-matrix of Gerstenhaber and Giaquinto. We analyze the structure of the elementary rime blocks and find, as a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly described in a rime form. We discuss then connections of the classical rime solutions with the Bezout operators. The Bezout operators satisfy the (non-)homogeneous associative classical Yang--Baxter equation which is related to the Rota-Baxter operators. We classify the rime Poisson brackets: they form a 3-dimensional pencil. A normal form of each individual member of the pencil depends on the discriminant of a certain quadratic polynomial. We also classify orderable quadratic rime associative algebras. For the standard Drinfeld-Jimbo solution, there is a choice of the multiparameters, for which it can be non-trivially rimed. However, not every Belavin-Drinfeld triple admits a choice of the multiparameters for which it can be rimed. We give a minimal example.Comment: 50 pages, typos correcte

On Inflation Rules for Mosseri-Sadoc Tilings

We give the inflation rules for the decorated Mosseri-Sadoc tiles in the projection class of tilings ${\cal T}^{(MS)}$. Dehn invariants related to the stone inflation of the Mosseri-Sadoc tiles provide eigenvectors of the inflation matrix with eigenvalues equal to $\tau = \frac{1+\sqrt{5}}{2}$ and $-\tau^{-1}$.Comment: LaTeX file, 4(3) pages + 7 figures (FIG1.gif, FIG2.gif,... FIH7.gif) and a style file (icqproc.sty

Reality in the Differential Calculus on q-euclidean Spaces

The nonlinear reality structure of the derivatives and the differentials for the euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.Comment: 10 page

Differential Calculus on h-Deformed Spaces

We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-type. In contrast to the $q$-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of ${\bf h}$-deformed differential operators $\operatorname{Diff}_{{\bf h},\sigma}(n)$ is labeled by a rational function $\sigma$ in $n$ variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings $\operatorname{Diff}_{{\bf h},\sigma}(n)$

Quantum dynamics of N=1N=1, D=4D=4 supergravity compensator

A new $N=1$ superfield theory in $D=4$ flat superspace is suggested. It describes dynamics of supergravity compensator and can be considered as a low-energy limit for $N=1$, $D=4$ superfield supergravity. The theory is shown to be renormalizable in infrared limit and infrared free. A quantum effective action is investigated in infrared domain

Classification of the GL(3) Quantum Matrix Groups

We define quantum matrix groups GL(3) by their coaction on appropriate quantum planes and the requirement that the Poincare series coincides with the classical one. It is shown that this implies the existence of a Yang-Baxter operator. Exploiting stronger equations arising at degree four of the algebra, we classify all quantum matrix groups GL(3). We find 26 classes of solutions, two of which do not admit a normal ordering. The corresponding R-matrices are given.Comment: 28 pages, Late