We construct a model of phantom energy using the graded Lie algebra SU(2/1). The negative kinetic energy of the phantom field emerges naturally from the graded Lie algebra, resulting in an equation of state with w < −1. The model also contains ordinary scalar fields and anti-commuting (Grassmann) vector fields which can be taken as two component dark matter. A potential term is generated for bo...

In this work, we describe a method to construct central polynomials for F -algebras where F is a field of characteristic zero. The main application deals with the T -prime algebras Mn(E), where E is the infinitedimensional Grassmann algebra over F , which play a fundamental role in the theory of PI-algebras. The method is based on the explicit decomposition of the group algebra FSn. AMS Classif...

A superspace formulation for the Batalin Vilkovisky formalism (also called fieldantifield quantization ) with extended BRST invariance (BRST and anti-BRST invariance ) for gauge theories with closed algebra is presented. In contrast to a recent formulation, where only BRST invariance holds off shell, two collective sets of fields are introduced and an off shell realization of the extended algeb...

In this paper, we investigate the entanglement of multi-partite Grassmannian coherent states (GCSs) described by Grassmann numbers for n > 2 degree of nilpotency. Choosing an appropriate weight function, we show that it is possible to construct some wellknown entangled pure states, consisting of GHZ, W, Bell, cluster type and bi-separable states, which are obtained by integrating over tensor pr...

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff’s matrix-tree theorem. Furthermore, we show tha...

The adjacency algebra of an association scheme is defined over an arbitrary field. In general, it is always semisimple over a field of characteristic zero but not always semisimple over a field of positive characteristic. The structures of adjacency algebras over fields of positive characteristic have not been sufficiently studied. In this paper, we consider the structures of adjacency algebras...

Exactly 125 years ago G. Peano introduced the modern concept of vectors in his 1888 book "Geometric Calculus According to the Ausdehnungslehre (Theory of Extension) of H. Grassmann". Unknown to Peano, the young British mathematician W. K. Cli ord (1846-1879) in his 1878 work "Applications of Grassmann's Extensive Algebra" had already 10 years earlier perfected Grassmann's algebra to the modern ...

Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators in H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. To do it we take the Jordan-Banach triple (or JB *-triple) approach which allows us to define a natural Levi-Civita connection on M by using algebraic tools. We identify the geodesics and the Riemann ...