Search

Let F be a field of characteristic 0, E be the unitary infinite dimensional Grassmann algebra over F and consider the algebra M1,1E with its natural Z2-grading. We describe the graded A-identities ...

Let E be the infinite dimensional Grassmann algebra over an infinite field of characteristic different from 2. Let E ( r 1 , … , r n ) ( v 1 , … , v n ) be a n-induced ( R , + ) -grading on E and l...

A ring R with center C is said to be centrally essential if the module RC is an essential extension of the module CC. We describe centrally essential exterior algebras of finitely generated free mo...

Let p be an odd prime number, F a field of characteristic zero, and let Ebe the unitary Grassmann algebra generated by the infinite-dimensionalF-vector space L. We determine the bases of the ℤp-graded identities.Moreover we compute the ℤp-graded codimension and cocharacter sequences for the algebra E endowed with any ℤp-grading such that L is a homogeneous subspace.

In [1], the authors computed the additive and multiplicative structure of HH*(A;A), where A is the n-th exterior algebra over a field. In this paper, we derive all their results using a different m...

We study the growth of the central polynomials for the algebras $G$ and $M_k(F)$, the infinite dimensional Grassmann algebra and the $k\times k$ matrices over a field $F$ of characteristic zero. In particular it follows that $M_k(F)$ satisfy many proper central polynomials.

This article gives a new matrix function named "twisted immanant," which can be regarded as an analogue of the immanant. This is defined for each self-conjugate partition through a "twisted" analogue of the irreducible character of the symmetric group. This twisted immanant has some interesting properties. For example, it satisfies Cauchy-Binet type formulas. Moreover it is closely related to the following results for matrices whose entries anticommute with each other: (i) the description of the invariants under the conjugations, and (ii) an analogue of the Cauchy identities for symmetric polynomials., Comment: 15 pages; revised version; to appear in Linear Multilinear Algebra

Since 1976, it is known from the paper by V. P. Belkin that the variety $\mathrm{RA_2}$ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains non-finitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann $\mathrm{RA_2}$-algebra of finite rank $r$ over a field $\mathcal{F}$, for $\mathrm{char}(\mathcal{F})\neq2$, is Spechtian iff $r=1$. We construct a non-finitely based variety $\mathfrak{M}$ generated by the Grassmann $\mathcal{V}$-algebra of rank $2$ of certain finitely based subvariety $\mathcal V\subset\mathrm{RA}_2$ over a field $\mathcal{F}$, for $\mathrm{char(\mathcal{F})}\neq2,3$, such that $\mathfrak{M}$ can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part., 20 pages

Let and be two non-commutative polynomials in disjoint sets of variables. An algebra is verbally prime if whenever is an identity for then either or is also an identity. As an analogue of this property, Regev proved that the verbally prime algebra of matrices over an infinite field has the following primeness property for central polynomials: whenever the product is a central polynomial for then both and are central polynomials. In this paper, we prove that over a field of characteristic zero, Regev’ s result holds for the verbally prime algebras and , where is the infinite-dimensional Grassmann algebra.

Some forms of Lie algebras of types E_6, E_7, and E_8 are constructed using the exterior cube of a rank 9 finitely generated projective module.

We study the exterior depth of an E-module and its exterior generic annihilator numbers. For the exterior depth of a squarefree E-module, we show how it relates to the symmetric depth of the corresponding S-module and classify those simplicial complexes having a particular exterior depth in terms of their exterior algebraic shifting. We define exterior annihilator numbers analogously to the annihilator numbers over the polynomial ring introduced by Trung [19] and Conca et al. [9]. In addition to a combinatorial interpretation of the annihilator numbers, we show how they are related to the symmetric Betti numbers and the Cartan–Betti numbers, respectively. We finally conclude with an example which shows that neither the symmetric nor the exterior generic annihilator numbers are minimal among the annihilator numbers with respect to a sequence.

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M n (F), M n (E) and M a,b (E) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand--Kirillov dimensions.

We describe the T-ideal of identities and the T-space of central polynomials for the unitary finite dimensional Grassmann algebra over a finite field.

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic 0. In this article we consider the algebra R of 2 × 2 matrices with entries in E and its subalgebra G, which is one of the minimal algebras of polynominal identity (PI) exponent 2. We compute firstly the Hilbert series of G and, as a consequence, we compute its cocharacter sequence. Then we find the Hilbert series of R, using the tool of proper Hilbert series, and we compute its cocharacter sequence. Finally we describe explicitely the ℤ2-graded cocharacters of R.

We describe the multilinear identities of the superalgebra M 2, 1(G) of matrices over the Grassmann algebra, in the minimal possible degree, which is 9.

We describe the T-ideal of identities and the T-space of central polynomials for the infinite dimensional unitary Grassmann algebra over a finite field.

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic zero, and consider L the F-vector space spanned by all generators of E. Let ϕ l be any fixed automorphism of E of order 2 such that L is an homogeneous subspace. Our goal is to finish the computation of the sequences of ϕ l -codimensions, by finding its exact value for the unique open case, that is, when the subspace of L corresponding to the eigenvalue 1 is finite-dimensional. As a consequence we get the ϕ-codimensions for a large amount of arbitrary automorphisms ϕ of E of order 2.

Let K be a field, char K = 0, and let E = E 0⊕ E 1 be the Grassmann algebra of infinite dimension over K, equipped with its natural ℤ2-grading. If G is a finite abelian group and R = ⨁ g∈G R (g) is a G-graded K-algebra, then the algebra R⊗ E can be G × ℤ2-graded by setting (R⊗ E)(g, i) = R (g) ⊗ E i . In this article we describe the graded central polynomials for the T-prime algebras M n (E)≅ M n (K)⊗ E. As a corollary we obtain the graded central polynomials for the algebras M a, b (E)⊗ E. As an application, we determine the ℤ2-graded identities and central polynomials for E⊗ E.

To do this, we show that it is possible to introduce simple ^-vectors?those of the form ax a A ak where ax,... , ak are vectors in some Rn?as geometric objects. In calculus or an introductory physics class, one sometimes hears a vector described as an equivalence class of directed line segments. It turns out that one can, in a similar way, define a simple /c-vector as an equivalence class of oriented parallelepipeds. We shall show that this geometric definition leads to the standard properties of exterior algebra. To illustrate the use and convenience of this machinery, we then generalize the triangle law of vector addition and the law of cosines to ft-simplexes and rederive the ft-dimensional versions of the parallelogram law and the Pythagorean theorem.

We establish results about algebraic shifting of simplicial complexes and use them to compare different shifting operations. In particular, we show that each shifting operation does not decrease the number of facets, and that the exterior shift is the best among the exterior shifting operations in the sense that it increases the number of facets the least. Methods of proof include Gr\"obner basis theory over the exterior algebra, Cartan homology, degree functions, and Alexander duality., Comment: 26 pages. Revised version. Proofs of theorems 2.10 (now 2.11) and 2.11 (now 2.12) corrected. Exposition improved with some reordering. Minor changes

Let ℱ be the family of graded ideals J in the exterior algebra E of a n-dimensional vector space over a field K such that e(E/J) = dim K (E/J) = e, indeg(E/J) = i and H E/J (i) = dim K (E/J) i are fixed integers. It is shown that there exists a unique lexsegment graded ideal J(n, e, i) ∊ ℱ whose Betti numbers give an upper bound for the Betti numbers of the ideals of ℱ. The authors continue the computation of upper bounds for the Betti numbers of graded ideals with given data started in Crupi and Utano (1999).

This essay is essentially a plea for recognizing the Aharonov–Bohm (AB) integral as part of a quantum superstructure of Maxwell theory counting units of flux. The Gauss–Ampere (GA) integral then is added as a counter of net charge as well as an exterior product of the two, suggested by Kiehn, as counter of units of action. De facto, the AB integral already has this status, yet how its prestatistical nature relates to standard statistics-based quantum mechanics needs delineation. For electromagnetism to claim this quantum superstructure as a legitimate member of its family it is unavoidable to take issue with the Copenhagen view of quantum mechanics as stated in the introductory paragraph of this article. The limiting mathematical traditions that have been standing in the way of such developments are briefly reviewed here. They pertain to the relations between local and global procedures as delineated in de Rham theory as well as recognizing certain metric-free aspects of electromagnetism as a requirement ...

The algebras M a, b (E) ⊗ E and M a+b (E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984–1987 (see Kemer 1991); other proofs of it were given by Regev (1990), and in several particular cases, by Di Vincenzo (1992), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M 1, 1(E) ⊗ E and M 2(E) when the base field is infinite and of characteristic p > 2. The algebra M a, a (E) ⊗ E satisfies certain graded identities that are not satisfied by M 2a (E). In another paper we proved that the algebras M 1, 1(E) and E ⊗ E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.

Let 𝕂 be a field of characteristic zero, and R be a G-graded 𝕂-algebra. We consider the algebra R ⊗ E, then deduce its G × ℤ2-graded polynomial identities starting from the G-graded polynomial identities of R. As a consequence, we describe a basis for the ℤ n × ℤ2-graded identities of the algebras M n (E). Moreover we give the graded cocharacter sequence of M 2(E), and show that M 2(E) is PI-equivalent to M 1,1(E) ⊗ E. This fact is a particular case of a more general result obtained by Kemer.

We describe an efficient way to use the Sn -module structure in the computation of the multilinear identities of degree n of a given algebra. The method was used to show that (where G is the Grassmann algebra) has identities of degree 8, but of no smaller degree. Explicit identities of degree 8 are given. It was also checked that and that has no identities of degree ≤ 8 apart from the consequences of the standard identity .

The notion of G-codimensions of G-identities of an algebra A was introduced by A. Giambruno and A. Regev in [1], where some basic properties of the sequence of G-codimensions were presented. In par...

We study how homogeneous ideals in the exterior algebra ? V over a finite-dimensional vector space V are minimally generated. In particular, we solve the following problems: Starting with an element pv of degree v, what is the maximum length l of a sequence pv, . . . , pv+l-l, with degpi = i, and such that pi is not in the ideal generated by pl, . . . , pi-l? What is the maximal possible number of minimal generators of degree d of a homogeneous ideal which does not contain all elements of degree d + 1? Our main tool is the Kruskal-Katona theorem.

Let W = ⊗n Vbe the nth tensor product of a vector space Vand let{e1,…ed} be a basis for V. Given a weakly increasing functionf,letW f⊗n Vbe the subspace of W: spanned by all eg=eg(1)⊗…⊗egnsuch that the monotone rearrangement of g isf. Let T fbe the set of all pairs [T, T1] of tableaux of shape λ=(λ1,…,λk), where T 1, with entries consisting of the multiset {f(1),…,f(n}, λ1+…+λk+n, is row-strict - column-weak, and T', with entries consisting of the set n⊆={1,…,n}, is row-strict - column-strict. For each a in the symmetric group on n, define a linear transformation P(rr) on (r)"K such that P(v1+-otimes;…⊗vn)=vσ−1(1)⊗…⊗Vσ−1n):. For each T 1, let σT 1 denote the permutation of n specfied in one-line notation by the entries of T 1 read left to right, top to bottom. For each T, let the tensor et be denned as a tensor product of elements of the Grassmann algebra by .In this paper, we prove (Theorem 3.4) that the set of tensors of the form P(σT 1)(eT ), [T, T 1]eTf , form a basis for Tf . By analogy with classica...

Let E be the exterior algebra of a finite dimensional vector space over a field K. Hilbert functions and Betti numbers of graded ideals i ⊂ E are studied. It is shown that there exists a unique graded ideal J ⊂ E such that for all graded ideals I ⊂ E generated in degree ≥ 2 such that one has for all i. Here is the i-th Betti number of a graded E-module M. The unique ideal J with the above property has “maximal” Hiibert function. Our result is the analogue of a theorem of Valla [10] which he proved for graded ideals in the polynomial ring. We further prove a similar inequality for the Bass numbers of E/I, and give a combinatorial application.

We show how the so called right and left determinants can be used in the solution of certain systems of linear equations over Lie niipotent rings, A close analogue of Cramer's rule is formulated for right linear equations over the Grassmann algebra.

We consider compound matrices and exterior products in order to generalize a fundamental relation between the eigenspace of a matrix A corresponding to a simple eigenvalue λ and certain minors of A − λI of order n − 1. By means of the Laplace Expansion Theorem, we show that if the geometric multiplicity of λ is k > 1, then the exterior product of k linearly eigenvectors corresponding to λ is uniquely (up to scalar multiples) determined by certain minors of A − λI of order n−k. Additional results are included on the nullspace of a compound matrix.

Bernstein-Gelfand-Gelfand showed that the derived category of coherent sheaves on the projective n-space Db(cohPn) is equivalent to the stable ategory of the category of Z-graded finite-dimensional modules over the exterior algebra. At the same time Beilinson gave a description of Db(cohPn) which can be interpreted in terms of tilting theory. We compare these two characterizations using Happel's description of the derived category of modules over a finite-dimensional algebra of fnite global dimension.

We prove that a commutative (resp. modified anticommutative) connected graded Hopf algebra over a commutative algebra over a field of characteristic zero is a symmetric algebra (resp. exterior algebra) if and only if it satisfies a certain “condistributive law”, expressed by a commutative diagram. Both results are obtained as a corollary of an analogous characterization of the symmetric superalgebra of a supermodule over a superring: The symmetric superalgebras over a superring R in characteristic zero are exactly the comodules over the symmetric superalgebra of R

Finite-dimensional vector spaces V V 2,…V k over a field F give rise to certain "enveloping" product spaces, viz., the tensor product and the Grassmann product An elegant means for defining these products is the use of the Universal Mapping Property in conjunction with the Cartesian product and k-iinear (or k-linear anti-symmetric) functionals. Now, if each Vi enjoys an inner product, then each inherit an inner product. On one hand, we present an over-view of so-called stand-alone (direct) results, i.e., those results concerning the scalar functions (elementary symmetric functions, inner product, determinant): on the other hand, we deal with applications (indirect results) which arise in complexity of computation, operator theory and numerical analysis.

If A, B are linear operators on a finite‐dimensional space it is known that AB and BA have the same characteristic polynomial. We show that this result can be proved very simply by the use of exterior algebra.

The theory of Procesi and Razmyslov on trace identities of matrices is (hook) generalized by generalizing the trace polynomials. In particular, we prove a hook version of an identity of Sn -characters. The generalized theory, which generalizes most of the earlier results, seem suitable for studying P.I. algebras of the form A ⨷ E where E is the Grassmann algebra.

The decentralized determinantal assignment problem (DDAP) is defined as the unifying description for the study of pole and zero assignment problems under decentralized output, state feedback (DOF, DSF) and decentralized ‘squaring down’ (DSD), respectively. DDAP is reduced to a linear problem of zero assignment of polynomial combinants and a multilinear problem of restricted decomposability of multivectors. The decentralization characteristic (DC) and the decentralized polynomial Grassmann representative )D — ℝ[s] — GR) of DDAP are defined. The fixed zero polynomial of DDAP is then determined as the zero polynomial of D— ℝ[s]—GR. The canonical D—ℝ[s]—GR, (CD—R[s]—GR) and the decentralized Plucker matrix (DPM) of DDAP are introduced and necessary conditions for arbitrary assignment of the non-fixed zeros are given in terms of the DPM. The family of strongly zero non-assignable (SNA( systems is defined, and for such systems the notion of the fixed zero is extended to that of the ‘almost fixed zero’ (AFZ). An...

Properties of the exterior algebra of a vector space are used to investigate the curvature operator of a Riemannian manifold. Induced inner products and linear maps are used to establish results about the Euler characteristic of a compact manifold. An open problem about the decomposition of operators on A 2 V is discussed. This problem arises in the study of the codimension needed for isometric embeddings. A new algebraic consequence of the first Bianchi identities is established.

This paper presents a motivated introduction to the concept of divergence of a vector field in Euclidean space at an advanced level. Use is made of some basic concepts from the exterior algebra of k‐forms and some from the calculus of differential forms.

To each finite-dimensional vector space E can canonically associate the space ΛE * ⊗ ΛE. This space has a natural inner product (bilinear, symmetric and non-degenerate) and two (algebraic) product structures. Identities relating these products should be important for understanding the structure of these algebras. Indeed such identities have been used, both to develop Mixed Exterior Algebra and to apply it. In the present paper, we obtain a new identity of this type and show that the identities mentioned above are consequences of it. In this sense, it may be regarded as basic to the subject.

To ease the student's passage from vector algebra to general linear algebra the author argues that the (non‐metrical) scalar triple product should be defined geometrically in terms of oriented volume and be taught beforethe (metrical) cross product.

The outgoing branches from a node of Coates' flow graph are interpreted as the elements of column vectors of the system matrix. The Grassmann algebra is used to calculate the exterior or outer product of these column vectors for evaluating the system determinant. Thus, the gain of a symbolic transfer function or the gain of a flow graph can be determined easily either by hand or by computers.