Duality for Ideals in the Grassmann Algebra

Abstract

Cataloged from PDF version of article.A duality is established between left and right ideals of a finite dimensional\ud Grassmann algebra such that if under the duality a left ideal I and a right ideal J\ud correspond then I is the left annihilator of J and J the right annihilator of I.\ud Another duality is established for two-sided ideals of the Grassmann algebra where\ud two ideals that correspond are annihilators of each other. The dual of the principal\ud ideal generated by an exterior 2-form is completely determined. Q 1996 Academic\ud Press, Inc

Annihilators of Principal Ideals in the Exterior Algebra

“…. , n s are all even and determine the generators of the ideal Ann(µ) under the assumption that the base field F is of characteristic 0, thus we obtain a generalization of the results of I. Dibag in [1]. In concluding we indicate that the restriction Char(F ) = 0 cannot be removed.…”

“…Clearly it satisfies the associativity B(ab, c) = B(a, bc) and therefore E(V ) is a Frobenius algebra. Thus the duality constructed in [1] is an immediate consequence of this fact and it is crucial for our proofs.…”

“…Each product involves 2r +t factors. Since 2r + t = s is the total number of the µ k each product is either 0 or ±µ 1 …”

“…Given a form µ and a form ω the determination of whether ω can be factorized as ω = µ ∧ τ by means of a number of exterior equations is proved to be a significant factorization problem in differential geometry (see Section 2.4 in [1]). Motivated by this factorization problem, in [1] I. Dibag determined annihilators of 2-vectors, by exhibiting its generators (as ideal).…”

“…Motivated by this factorization problem, in [1] I. Dibag determined annihilators of 2-vectors, by exhibiting its generators (as ideal). This is accomplished by setting up a duality between left and right ideals of the exterior algebra, that is to say by using Frobeniusean property of exterior algebras.…”