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In this article, the structure of the Clifford-Weyl superalgebras and their associated Lie superalgebras will be investigated. These superalgebras have a natural supersymmetric inner product which is invariant under their Lie superalgebra structures. The Clifford-Weyl superalgebras can be realized as tensor product of the algebra of alternating and symmetric tensors respectively, on the even and odd parts of their underlying superspace. For Physical applications in elementaryparticles, we add star structures to these algebras and investigate the basic relations. Ortho-symplectic Lie algebras are naturally present in these algebras and their representations on these algebras can be described easily.

Let K be a field, E the exterior algebra of a finite dimensional K-vector space, and F a finitely generated graded free E-module with homogeneous basis g1, . . ., gr such that deg g1 ≤ deg g2 ≤ · · · ≤ deg gr. We characterize the Hilbert functions of graded E–modules of the type F/M, with M graded submodule of F. The existence of a unique lexicographic submodule of F with the same Hilbert function as M plays a crucial role.

In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler–Lagrange equations.

The basic concepts of exterior calculus for space−time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space−time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms.