A Clifford algebra has three major multiplications: inner product, outer product and geometric product. Accordingly, the same Clifford algebra has three versions: Clifford vector algebra, which features on inner products and outer products of multivectors; Clifford bracket algebra, which features on pseudoscalars and inner products of vectors; Clifford geometric algebra, which features on geome...

The notions of N = 1 Neveu-Schwarz vertex operator superal-gebra over a Grassmann algebra and with odd formal variables and of N = 1 Neveu-Schwarz vertex operator superalgebra over a Grass-mann algebra and without odd formal variables are introduced, and we show that the respective categories of such objects are isomorphic. The weak supercommutativity and weak associativity properties for an N ...

This paper is a review of monopoles, lowest Landau level, fuzzy spheres, and their mutual relations. The Hopf maps of division algebras provide a prototype relation between monopoles and fuzzy spheres. Generalization of complex numbers to Clifford algebra is exactly analogous to generalization of fuzzy two-spheres to higher dimensional fuzzy spheres. Higher dimensional fuzzy spheres have an int...

We discuss a new approach to putting supersymmetric theories on the lattice. The basic idea is to start from a twisted formulation of the underlying supersymmetric theory in which the fermions are represented as grassmann valued antisymmetric tensor fields. The original supersymmetry algebra is replaced by a twisted algebra which contains a scalar nilpotent supercharge Q. Furthermore the action...

Superfield expansions over four-dimensional graded spacetime (x µ , θ ν), with Minkowski coordinates x extended by vector Grassmann variables θ, are investigated. By appropriate identification of the physical Lorentz algebra in the even and odd parts of the superfield, a typology of 'schizofields' containing both integer and half-integer spin fields is established. For two of these types, ident...

In this paper, we use Clifford algebra and the spinor calculus to study the complex structures on Euclidean space R8 and the spheres S4, S6. By the spin representation of G(2, 8) ⊂ Spin(8) we show that the Grassmann manifold G(2, 8) can be looked as the set of orthogonal complex structures on R8. In this way, we show that G(2, 8) and CP 3 can be looked as twistor spaces of S6 and S4 respectivel...

In this paper we introduce Symplectic Grassmann codes, in analogy to ordinary Grassmann codes and Orthogonal Grassmann codes, as projective codes defined by symplectic Grassmannians. Lagrangian–Grassmannian codes are a special class of Symplectic Grassmann codes. We describe all the parameters of line Symplectic Grassmann codes and we provide the full weight enumerator for the Lagrangian–Grassm...

Let Θ be a point in R. We are concerned with the approximation to Θ by rational linear subvarieties of dimension d for 0 ≤ d ≤ n−1. To that purpose, we introduce various convex bodies in the Grassmann algebra Λ(R). It turns out that our convex bodies in degree d are the d-th compound, in the sense of Mahler, of convex bodies in degree one. A dual formulation is also given. This approach enables...

Let j/ be a unital C*-algebra. We describe K-skeleton factorizations of all invertible operators on a Hubert C*-module %^ , in particular on «^ = I2 , with the Fredholm index as an invariant. We then outline the isomorphisms K0(s/) =i TC2k([p]0) 3 n2k(GLp(s/)) and Kx(s*) & ä2*+i([Pjo) = n2k+\(GLpr(s?)) for k > 0 , where [p]o denotes the class of all compact perturbations of a projection p in th...