Minimal generators of annihilators of even neat elements in the exterior algebra footnotesize

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, when we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras Cl(p,q) can be embedded in Cl(p+1,q+1), which is shown to be exactly the extended Clifford algebra. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms.

Introduction The free algebra on a set, and the tensor and exterior algebras of a vector space are constructed. Emphasis is placed on the exterior algebra of a finite dimensional vector space and its applications to determinants. Afterwards an alternate independent development of the exterior algebra is given as the quotient of a tensor algebra. Why is the exterior and to a lesser extent the tensor algebra given such prominence in this chapter as opposed to other algebras? The exterior algebra appears in differential geometry not only as the algebra of differential forms but also from skew symmetric tensors. There seems to be an abundant supply of the latter perhaps because the Lie product [ x , y ] = xy – yx is skew symmetric in x and y , and various Lie operations (e.g. the Lie derivative) and Lie algebras appear unavoidably in differential geometry and in physics as well. Some constructions and concepts in physics can be formulated more precisely, more easily, and be better understood if one has the tensor and exterior algebras at one's disposal. There is a lot of interesting and useful material here that is not covered in this chapter, which hopefully will serve as an introduction and invitation to further study. This chapter takes place in the category of algebras with identity over a fixed ground field F and identity preserving homomorphisms. The identity element of a subalgebra is required to be the identity of the big algebra containing it.

Various computations in Grassmann and Clifford algebras can be performed with a Maple package CLIFFORD. It can solve algebraic equations when searching for general elements satisfying certain conditions, solve an eigenvalue problem for a Clifford number, and find its minimal polynomial. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. It uses standard (undotted) Grassmann basis in Cl(Q) but when the antisymmetric part of B is non zero, it can also compute in a dotted Grassmann basis. Some examples of computations are discussed.

This article discusses invariant theories in some exterior algebras, which are closely related to Amitsur–Levitzki type theorems.First we consider the exterior algebra on the vector space of square matrices of size n, and look at the invariants under conjugations. We see that the algebra of these invariants is isomorphic to the exterior algebra on an n-dimensional vector space. Moreover we give a Cayley–Hamilton type theorem for these invariants (the anticommutative version of the Cayley–Hamilton theorem). This Cayley–Hamilton type theorem can also be regarded as a refinement of the Amitsur–Levitzki theorem.We discuss two more Amitsur–Levitzki type theorems related to invariant theories in exterior algebras. One is a famous Amitsur–Levitzki type theorem due to Kostant and Rowen, and this is related to O(V)-invariants in Λ(Λ2(V)). The other is a new Amitsur–Levitzki type theorem, and this is related to GL(V)-invariants in Λ(Λ2(V)⊕S2(V⁎)).

formula omitted]-graded polynomial identities of the Grassmann algebra

Automorphisms and superalgebra structures on the Grassmann algebra

Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations Differentiable Manifolds: 1. Definitions 2. Differential maps 3. Sard's theorem 4. Partitions of unity, approximation theorems 5. The tangent space 6. The principal bundle 7. The tensor bundles 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$ 2. Chains and integration 3. Integration of densities 4. $0$ and $n$-dimensional cohomology, degree 5. Frobenius' theorem 6. Darboux's theorem 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations 2. Necessary conditions 3. Conservation laws 4. Sufficient conditions 5. Conjugate and focal points, Jacobi's condition 6. The Riemannian case 7. Completeness 8. Isometries Lie Groups: 1. Definitions 2. The invariant forms and the Lie algebra 3. Normal coordinates, exponential map 4. Closed subgroups 5. Invariant metrics 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space 2. The equations of structure of a submanifold 3. The equations of structure of a Riemann manifold 4. Curves in Euclidean space 5. The second fundamental form 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections 2. $G$-structures 3. Prolongations 4. Structures of finite type 5. Connections on $G$-structures 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.

The main purpose of this paper is to express the arithmetic notion of genus which appears in E. Cartan's Theory of exterior differential equations [2] in terms of ideal theory. Accordingly Sections 1, 2, and 3 of this paper are devoted to a discussion of ideal theory in Grassmann algebras. Let V be an n dimensional vector space and let V* be its dual space. Let A(V) and A(V*) denote the Grassmann algebra over V and V* respectively. Now A(V) and A(V*) are dually paired to the reals. Let W be a subspace of V and let A(W) be the Grassmann algebra over W. Then A(W) C A(V). We wish to study the annihilator space in A(V*) of A(W). In Theorem 2.3 we show that this is exactly the homogeneous ideal generated in A(V*) by the annihilator of W. An ideal generated by a subspace of V* will be called a one generated ideal. It should be remarked that the integral elements of E. Cartan are precisely Grassmann algebras generated by subspaces of V. We also define a concept of minimal ideal, see Definition 3.1, for a given ideal W and prove that under certain conditions on 9a the dimension of the covector space which generates a minimal ideal is an invariant of the ideal Wf. We call this number the dimension of 2f when it exists. Sections 4, 5, and 6 are devoted to applications of the above techniques to the study of the characteristic vector space, the genus of a differential ideal and the problem of prolongation, respectively. We prove that the characteristic vector space is contained in the intersection of the annihilator spaces of the minimal ideals for the given ideal W. We also find a condition for these two spaces to coincide. Theorem 5.1 shows that for one and two generated ideals Xf, the genus at a point is either one or the dimension of W. In Theorem 5.2 we give a necessary and sufficient condition for the genus to be one. In Section 6 we give an invariant formulation of the process of prolongation and in Theorem 6.2 we obtain a characterization of certain integral ideals in the prolonged system. We will assume in presenting the material of this paper that the reader is familiar with [1] and [2]. The author would like to acknowledge the fact that his interest in this subject stems from a course given by Professor S. S. Chern at the University of Chicago in 1952-1953 and that much of the exposition in paragraphs 5 and 6 follow the ideas in [3].

The chapter begins with notion of the Plucker coordinates of a subspace in a vector space. Then the Plucker relations are derived, and the Grassmann varieties are described. Then an exterior product of vectors is defined, and the connection between the exterior product and Plucker coordinates is explained: the Plucker relations give necessary and sufficient conditions for an m−vector to be represented as an exterior product of m vectors of the initial vector space. The properties of an exterior product are investigated in greater detail; the notion of exterior algebra is introduced and discussed. Finally, several applications of the obtained theoretical results are considered, for instance, Laplace’s formula for determinant of a square matrix and the Cauchy–Binet formula for the determinant of the matrix product are proved using the exterior product.

Let E be the Grassmann algebra of an infinite-dimensional vector space L over a field of characteristic zero. In this paper, we study the -gradings on E having the form , in which each element of a basis of L has -degree , or . We provide a criterion for the support of this structure to coincide with a subgroup of the group , and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in Brandão A, Fidelis C, Guimarães A. -gradings of full support on the Grassmann algebra. J Algebra. 2022;601:332–353. DOI:10.1016/j.jalgebra.2022.03.014. See also in arXiv preprint, arXiv:2009.01870v1, 2020].

Clifford algebras are associative algebras similar to Grassmann algebras, which we have more or less met in chapter 7. Grassmann algebras are just the algebras of anti-symmetric tensors. Originally, both were developed for facilitating geometrical computations, and indeed Grassmann algebras have been recently rediscovered by computer scientists for performing calculations in computational geometry. In fact, as we will see a little later, the only difference between Clifford algebras and Grassmann algebras is a slight difference in the algebraic structure. The reason for the utility of these algebras is that they contain several representations of the orthogonal group. That is, given a vector space ℝn we can construct the corresponding Clifford or Grassmann algebra containing various representations of O (n). In particular, we have already seen that the Grassmann algebra of anti-symmetric tensors on ℝn contains all the representations ∧kℝn corresponding to the action of O (n) on k-planes. Clifford algebras have an even richer structure and are also known as geometric algebras. They have even been proposed as “a unified language for mathematics and physics”; see Hestenes and Sobczyk [37].KeywordsDouble CoverClifford AlgebraGeometric AlgebraDual VectorGrassmann AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a complex geometry resulting from a transitive action of an appropriate algebraic group, yielding a compact model submanifold for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2 torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. The cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the stable range as the stable homotopy groups of the associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we obtain a class of formal linear combinations of exceptional orbit hypersurfaces which have Milnor fibers which are homotopy equivalent to joins of the compact model submanifolds.

We consider the topology for a class of hypersurfaces with highly nonisolated singularities which arise as ‘exceptional orbit varieties’ of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements, and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse-type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a ‘complex geometry’ resulting from a transitive action of an appropriate algebraic group, yielding a compact ‘model submanifold’ for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2-torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. Unlike isolated singularities, the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the ‘stable range’ as the stable homotopy groups of the associated infinite-dimensional symmetric spaces. Lastly, we combine the preceding with a Theorem of Oka to obtain a class of ‘formal linear combinations’ of exceptional orbit hypersurfaces which have Milnor fibers that are homotopy equivalent to joins of the compact model submanifolds. It follows that Milnor fibers for all of these hypersurfaces are essentially never homotopy equivalent to bouquets of spheres (even allowing differing dimensions).

In this chapter, we recall some basic definitions and facts concerning exterior products (which are essential in the discussion of differential forms in Sect. 9.5) and tensor products (which are essential in the discussion of holomorphic line bundles in Chap. 3). In this book, we mostly consider exterior and tensor products in vector spaces of dimension 1 or 2.