The ^-differential calculus for the g-Minkowski space is developed. The algebra of the ^-derivatives with the g-Lorentz generators is found giving thê -deformation of the. Poincare algebra. The reality structure of the g-Poincare algebra is given. The reality structure of the ^-differentials is also found. The real Laplacian is constructed. Finally the comultiplication, counit and antipode for thê -Poincare algebra are obtained making it a Hopf algebra.

We find a principle of harmonic analyticity underlying the quaternionic (quaternion-Kähler) geometry and solve the differential constraints which define this geometry. To this end the original 4n-dimensional quaternionic manifold is extended to a biharmonic space. The latter includes additional harmonic coordinates associated with both the tangent local Sp(1) group and an extra rigid SU (2) group rotating the complex structures. Then the constraints can be rewritten as integrability conditions for the existence of an analytic subspace in the bi-harmonic space and solved in terms of two unconstrained potentials on the analytic subspace. Geometrically, the potentials have the meaning of vielbeins associated with the harmonic coordinates. We also establish a one-to-one correspondence between the quaternionic spaces and off-shell N = 2 supersymmetric sigma-models coupled to N = 2 supergravity. The general N = 2 sigma-model Lagrangian when written in the harmonic superspace is composed of the quaternionic potentials. Coordinates of the analytic subspace are identified with superfields describing N = 2 matter hypermultiplets and a compensating hypermultiplet of N = 2 supergravity. As an illustration we present the potentials for the symmetric quaternionic spaces. † On leave from the Laboratory of Theoretical Physics, JINR, Dubna, Head Post Office,

Non-polynomial baxterized solutions of reflection equations associated with affine Hecke and affine Birman-Murakami-Wenzl algebras are found. Relations to integrable spin chain models with nontrivial boundary conditions are discussed.

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.