DOCSLIB.ORG

1 Review of Multilinear Algebra

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Review of Multilinear Algebra Review of Multilinear Algebra by Min Ru 1 Review of Multilinear Algebra We mainly deal in the following with finite-dimensional vector spaces. Let V be an m- m dimensional real vector space. If we choose a basis {ei}i=1, V is isomorphic to the Euclidean space Rm by assigining its components to eact element of V . Now we review briefly some methods by which produce new vector spaces out of the given spaces. • Dual space. V ∗ := {α : V → R; α is a linear map} has the structure of an m- i dimensional vector space and is called the duea space of V . We define e (ej) := δij, i m ∗ m then {e }i=1 forms a basis of V which is called the dual basis of {ei}i=1. • Tensor product. Let V and W be vector spaces of dimension m and n. Then the space Hom(V, W ) := {φ : V → W ; φ is a linear map} has the structure of an mn- dimensional vector space. Hom(V ∗,W ), also denoted by V ⊗ W , is called the tensor product of V and W . For v ∈ V, w ∈ W we define v ⊗ w ∈ V ⊗ W by v ⊗ w(v∗) := v∗(v)w. Then any element of V ⊗ W may be expressed by a linear combination of elements of the form v ⊗ w, and, in fact, {ei ⊗ fj}1≤i≤m,1≤j≤n forms a basis of V ⊗ W . It is important to note that V ⊗ W is isomorphic to the vector space {φ : V ∗ × W ∗ → R; φ is a bilinear map} by assigning to v ⊗ w the bilinear map: (v∗, w∗) 7→ w∗(v ⊗ w(v∗)) = v∗(v)w∗(w) ∈ R. 1 • Tensor space. For a vector space V , we define the tensor space of type (r, s) of V , which is denoted by r ∗ ∗ Ts (V ) = V ⊗ · · · ⊗ V ⊗ V ⊗ · · · ⊗ V , as the vector space {φ : V ∗ × · · · × V ∗ × V × · · · × V → R; φ is a multilinear map}, where the first products for V is taken r times, and the second products for V ∗ is taken s times. Its element is called tensors of type (r, s). ∗ ∗ If xi ∈ V (1 ≤ i ≤ r), yj ∈ V (1 ≤ j ≤ rs) are given, then we get an (r, s)-tensor by the following formula: ∗ ∗ ∗ ∗ Y ∗ ∗ x1 ⊗ · · · ⊗ xr ⊗ y1 ⊗ · · · ⊗ ys (x1, . , xr, y1, . , ys) := xi (xi)yj (yj). i,j j1 js r Then, we easily see that {ei1 ⊗ · · · ⊗ eir ⊗ e ⊗ · · · ⊗ e } forms a basis for Ts (V ) and r r+s r dim Ts (V ) = m . Thus, every t ∈ Ts (V ) may be expressed as X i1,...,ir j1 js t = tj1...,js ei1 ⊗ · · · ⊗ eir ⊗ e ⊗ · · · ⊗ e . i1,...,ir,j1...,js r ∗ ∼ r ∗ r+r0 ∼ We note that we have canonical isomorphisms Ts (V ) = Ts (V ) and Ts+s0 (V ) = r r0 Ts (V ) ⊗ Ts0 (V ). • Exterior algebra. We recall that the vector space k ^(V ) := {α : V × · · · V → R; α is a skew-symmetric k-linear map}, where the product is taken k-times. Vk(V ) is called the k-th exterior product of V ∗ and its elements are called the k-forms. Here α is said to be skew-symmetric if for any 2 permutation σ of {1, . , k} we have α(xσ(1), ··· , xσ(k)) = sgnσ · α(x1, ··· , xk), where sgnσ denotes the sign of a permutation σ. For instance, we define ∗ ∗ ∗ x1 ∧ · · · ∧ xk(x1, . , xk) := det(xi (xj)). ∗ ∗ Vk ∗ ∗ Then we easily check that x1 ∧ · · · ∧ xr ∈ (V ) and that xσ(1) ∧ · · · ∧ x σ(k) = ∗ ∗ i1 ik Vk sgnσx1 ∧ · · · ∧ xk. Then {e ∧ · · · ∧ e , i1 < ··· < ik} forms a basis for (V ) and Vk (m+k)! V0 V1 ∗ Vk dim (V ) = m!k! . In particular, we have (V ) = R, (V ) = V , (V ) = {0} if k > m. Further we define for α ∈ Vk(V ) and β ∈ Vl(V ) their exterior product α ∧ β ∈ Vk+l(V ) by 1 X α ∧ β(x1, . , xk+l) := (sgnσ)α(xσ(1), . , xσ(k))β(xσ(k+1), . , xσ(k+l)). k!l! σ kl V∗ Lm Vk Note that α ∧ β = (−1) β ∧ α and (V ) := k=0 (V ) has the structure of an algebra with respect to “∧”. Now in the same manner, we may construct ^(V ) := {ξ : V ∗ × · · · V ∗ → R; ξ is a skew-symmetric k-linear map}, k where the product is taken k-times. Then {ei1 ∧ · · · ∧ eik , i1 < ··· < ik} forms a basis V V V for k(V ), and we may considet the exterior product ξ ∧ η ∈ k+l(V ) of ξ ∈ k(V ) V and η ∈ l(V ) as above. 3.
Load more

Recommended publications
  • Multilinear Algebra and Applications July 15, 2014
    Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3.
    [Show full text]
  • Causalx: Causal Explanations and Block Multilinear Factor Analysis
    To appear: Proc. of the 2020 25th International Conference on Pattern Recognition (ICPR 2020) Milan, Italy, Jan. 10-15, 2021. CausalX: Causal eXplanations and Block Multilinear Factor Analysis M. Alex O. Vasilescu1;2 Eric Kim2;1 Xiao S. Zeng2 [email protected] [email protected] [email protected] 1Tensor Vision Technologies, Los Angeles, California 2Department of Computer Science,University of California, Los Angeles Abstract—By adhering to the dictum, “No causation without I. INTRODUCTION:PROBLEM DEFINITION manipulation (treatment, intervention)”, cause and effect data Developing causal explanations for correct results or for failures analysis represents changes in observed data in terms of changes from mathematical equations and data is important in developing in the causal factors. When causal factors are not amenable for a trustworthy artificial intelligence, and retaining public trust. active manipulation in the real world due to current technological limitations or ethical considerations, a counterfactual approach Causal explanations are germane to the “right to an explanation” performs an intervention on the model of data formation. In the statute [15], [13] i.e., to data driven decisions, such as those case of object representation or activity (temporal object) rep- that rely on images. Computer graphics and computer vision resentation, varying object parts is generally unfeasible whether problems, also known as forward and inverse imaging problems, they be spatial and/or temporal. Multilinear algebra, the algebra have been cast as causal inference questions [40], [42] consistent of higher order tensors, is a suitable and transparent framework for disentangling the causal factors of data formation. Learning a with Donald Rubin’s quantitative definition of causality, where part-based intrinsic causal factor representations in a multilinear “A causes B” means “the effect of A is B”, a measurable framework requires applying a set of interventions on a part- and experimentally repeatable quantity [14], [17].
    [Show full text]
  • 28. Exterior Powers
    28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1.
    [Show full text]
  • Multilinear Algebra in Data Analysis: Tensors, Symmetric Tensors, Nonnegative Tensors
    Multilinear Algebra in Data Analysis: tensors, symmetric tensors, nonnegative tensors Lek-Heng Lim Stanford University Workshop on Algorithms for Modern Massive Datasets Stanford, CA June 21–24, 2006 Thanks: G. Carlsson, L. De Lathauwer, J.M. Landsberg, M. Mahoney, L. Qi, B. Sturmfels; Collaborators: P. Comon, V. de Silva, P. Drineas, G. Golub References http://www-sccm.stanford.edu/nf-publications-tech.html [CGLM2] P. Comon, G. Golub, L.-H. Lim, and B. Mourrain, “Symmetric tensors and symmetric tensor rank,” SCCM Tech. Rep., 06-02, 2006. [CGLM1] P. Comon, B. Mourrain, L.-H. Lim, and G.H. Golub, “Genericity and rank deficiency of high order symmetric tensors,” Proc. IEEE Int. Con- ference on Acoustics, Speech, and Signal Processing (ICASSP), 31 (2006), no. 3, pp. 125–128. [dSL] V. de Silva and L.-H. Lim, “Tensor rank and the ill-posedness of the best low-rank approximation problem,” SCCM Tech. Rep., 06-06 (2006). [GL] G. Golub and L.-H. Lim, “Nonnegative decomposition and approximation of nonnegative matrices and tensors,” SCCM Tech. Rep., 06-01 (2006), forthcoming. [L] L.-H. Lim, “Singular values and eigenvalues of tensors: a variational approach,” Proc. IEEE Int. Workshop on Computational Advances in Multi- Sensor Adaptive Processing (CAMSAP), 1 (2005), pp. 129–132. 2 What is not a tensor, I • What is a vector? – Mathematician: An element of a vector space. – Physicist: “What kind of physical quantities can be rep- resented by vectors?” Answer: Once a basis is chosen, an n-dimensional vector is something that is represented by n real numbers only if those real numbers transform themselves as expected (ie.
    [Show full text]
  • Multilinear Algebra
    Appendix A Multilinear Algebra This chapter presents concepts from multilinear algebra based on the basic properties of finite dimensional vector spaces and linear maps. The primary aim of the chapter is to give a concise introduction to alternating tensors which are necessary to define differential forms on manifolds. Many of the stated definitions and propositions can be found in Lee [1], Chaps. 11, 12 and 14. Some definitions and propositions are complemented by short and simple examples. First, in Sect. A.1 dual and bidual vector spaces are discussed. Subsequently, in Sects. A.2–A.4, tensors and alternating tensors together with operations such as the tensor and wedge product are introduced. Lastly, in Sect. A.5, the concepts which are necessary to introduce the wedge product are summarized in eight steps. A.1 The Dual Space Let V be a real vector space of finite dimension dim V = n.Let(e1,...,en) be a basis of V . Then every v ∈ V can be uniquely represented as a linear combination i v = v ei , (A.1) where summation convention over repeated indices is applied. The coefficients vi ∈ R arereferredtoascomponents of the vector v. Throughout the whole chapter, only finite dimensional real vector spaces, typically denoted by V , are treated. When not stated differently, summation convention is applied. Definition A.1 (Dual Space)Thedual space of V is the set of real-valued linear functionals ∗ V := {ω : V → R : ω linear} . (A.2) The elements of the dual space V ∗ are called linear forms on V . © Springer International Publishing Switzerland 2015 123 S.R.
    [Show full text]
  • CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V
    CHAPTER 2 Clifford algebras 1. Exterior algebras 1.1. Definition. For any vector space V over a field K, let T (V ) = k k k Z T (V ) be the tensor algebra, with T (V ) = V V the k-fold tensor∈ product. The quotient of T (V ) by the two-sided⊗···⊗ ideal (V ) generated byL all v w + w v is the exterior algebra, denoted (V ).I The product in (V ) is usually⊗ denoted⊗ α α , although we will frequently∧ omit the wedge ∧ 1 ∧ 2 sign and just write α1α2. Since (V ) is a graded ideal, the exterior algebra inherits a grading I (V )= k(V ) ∧ ∧ k Z M∈ where k(V ) is the image of T k(V ) under the quotient map. Clearly, 0(V )∧ = K and 1(V ) = V so that we can think of V as a subspace of ∧(V ). We may thus∧ think of (V ) as the associative algebra linearly gener- ated∧ by V , subject to the relations∧ vw + wv = 0. We will write φ = k if φ k(V ). The exterior algebra is commutative | | ∈∧ (in the graded sense). That is, for φ k1 (V ) and φ k2 (V ), 1 ∈∧ 2 ∈∧ [φ , φ ] := φ φ + ( 1)k1k2 φ φ = 0. 1 2 1 2 − 2 1 k If V has finite dimension, with basis e1,...,en, the space (V ) has basis ∧ e = e e I i1 · · · ik for all ordered subsets I = i1,...,ik of 1,...,n . (If k = 0, we put { } k { n } e = 1.) In particular, we see that dim (V )= k , and ∅ ∧ n n dim (V )= = 2n.
    [Show full text]
  • Mathematics 552 Algebra II Spring 2017 the Exterior Algebra of K
    Mathematics 552 Algebra II Spring 2017 The exterior algebra of K-modules and the Cauchy-Binet formula Let K be a commutative ring, and let M be a K-module. Recall that the tensor ⊗i algebra T (M) = ⊕i=0M is an associative K-algebra and that if M is free of finite rank ⊗k m with basis v1; : : : ; vm then T (M) is a graded free K module with basis for M given ⊗k by the collection of vi1 ⊗ vi2 ⊗ · · · vik for 1 ≤ it ≤ m. Hence M is a free K-module of rank mk. We checked in class that the tensor algebra was functorial, in that when f : M ! N is a K-module homomorphism, then the induced map f ⊗i : M ⊗i ! N ⊗i gives a homomorphism of K-algebras T (f): T (M) ! T (N) and that this gives a functor from K-modules to K-algebras. The ideal I ⊂ T (M) generated by elements of the form x ⊗ x for x 2 M enables us to construct an algebra Λ(M) = T (M)=I. Since (v+w)⊗(v+w) = v⊗v+v⊗w+w⊗v+w⊗w we have that (modulo I) v ⊗ w = −w ⊗ v. The image of M ⊗i in Λ(M) is denoted Λi(M), the i-th exterior power of M. The product in Λ(M) derived from the product on T (M) is usually called the wedge product : v ^ w is the image of v ⊗ w in T (M)=I. Since making tensor algebra of modules is functorial, so is making the exterior algebra.
    [Show full text]
  • Multilinear Algebra for Visual Geometry
    multilinear algebra for visual geometry Alberto Ruiz http://dis.um.es/~alberto DIS, University of Murcia MVIGRO workshop Berlin, june 28th 2013 contents 1. linear spaces 2. tensor diagrams 3. Grassmann algebra 4. multiview geometry 5. tensor equations intro diagrams multiview equations linear algebra the most important computational tool: linear = solvable nonlinear problems: linearization + iteration multilinear problems are easy (' linear) visual geometry is multilinear (' solved) intro diagrams multiview equations linear spaces I objects are represented by arrays of coordinates (depending on chosen basis, with proper rules of transformation) I linear transformations are extremely simple (just multiply and add), and invertible in closed form (in least squares sense) I duality: linear transformations are also a linear space intro diagrams multiview equations linear spaces objects are linear combinations of other elements: i vector x has coordinates x in basis B = feig X i x = x ei i Einstein's convention: i x = x ei intro diagrams multiview equations change of basis 1 1 0 0 c1 c2 3 1 0 i [e1; e2] = [e1; e2]C; C = 2 2 = ej = cj ei c1 c2 1 2 x1 x1 x01 x = x1e + x2e = [e ; e ] = [e ; e ]C C−1 = [e0 ; e0 ] 1 2 1 2 x2 1 2 x2 1 2 x02 | 0{z 0 } [e1;e2] | {z } 2x013 4x025 intro diagrams multiview equations covariance and contravariance if the basis transforms according to C: 0 0 [e1; e2] = [e1; e2]C vector coordinates transform according to C−1: x01 x1 = C−1 x02 x2 in the previous example: 7 3 1 2 2 0;4 −0;2 7 = ; = 4 1 2 1 1 −0;2 0;6 4 intro
    [Show full text]
  • Lecture Notes on Linear and Multilinear Algebra 2301-610
    Lecture Notes on Linear and Multilinear Algebra 2301-610 Wicharn Lewkeeratiyutkul Department of Mathematics and Computer Science Faculty of Science Chulalongkorn University August 2014 Contents Preface iii 1 Vector Spaces 1 1.1 Vector Spaces and Subspaces . 2 1.2 Basis and Dimension . 10 1.3 Linear Maps . 18 1.4 Matrix Representation . 32 1.5 Change of Bases . 42 1.6 Sums and Direct Sums . 48 1.7 Quotient Spaces . 61 1.8 Dual Spaces . 65 2 Multilinear Algebra 73 2.1 Free Vector Spaces . 73 2.2 Multilinear Maps and Tensor Products . 78 2.3 Determinants . 95 2.4 Exterior Products . 101 3 Canonical Forms 107 3.1 Polynomials . 107 3.2 Diagonalization . 115 3.3 Minimal Polynomial . 128 3.4 Jordan Canonical Forms . 141 i ii CONTENTS 4 Inner Product Spaces 161 4.1 Bilinear and Sesquilinear Forms . 161 4.2 Inner Product Spaces . 167 4.3 Operators on Inner Product Spaces . 180 4.4 Spectral Theorem . 190 Bibliography 203 Preface This book grew out of the lecture notes for the course 2301-610 Linear and Multilinaer Algebra given at the Deparment of Mathematics, Faculty of Science, Chulalongkorn University that I have taught in the past 5 years. Linear Algebra is one of the most important subjects in Mathematics, with numerous applications in pure and applied sciences. A more theoretical linear algebra course will emphasize on linear maps between vector spaces, while an applied-oriented course will mainly work with matrices. Matrices have an ad- vantage of being easier to compute, while it is easier to establish the results by working with linear maps.
    [Show full text]
  • Numerical Multilinear Algebra I
    Numerical Multilinear Algebra I Lek-Heng Lim University of California, Berkeley January 5{7, 2009 L.-H. Lim (ICM Lecture) Numerical Multilinear Algebra I January 5{7, 2009 1 / 55 Hope Past 50 years, Numerical Linear Algebra played indispensable role in the statistical analysis of two-way data, the numerical solution of partial differential equations arising from vector fields, the numerical solution of second-order optimization methods. Next step | development of Numerical Multilinear Algebra for the statistical analysis of multi-way data, the numerical solution of partial differential equations arising from tensor fields, the numerical solution of higher-order optimization methods. L.-H. Lim (ICM Lecture) Numerical Multilinear Algebra I January 5{7, 2009 2 / 55 DARPA mathematical challenge eight One of the twenty three mathematical challenges announced at DARPA Tech 2007. Problem Beyond convex optimization: can linear algebra be replaced by algebraic geometry in a systematic way? Algebraic geometry in a slogan: polynomials are to algebraic geometry what matrices are to linear algebra. Polynomial f 2 R[x1;:::; xn] of degree d can be expressed as > > f (x) = a0 + a1 x + x A2x + A3(x; x; x) + ··· + Ad (x;:::; x): n n×n n×n×n n×···×n a0 2 R; a1 2 R ; A2 2 R ; A3 2 R ;:::; Ad 2 R . Numerical linear algebra: d = 2. Numerical multilinear algebra: d > 2. L.-H. Lim (ICM Lecture) Numerical Multilinear Algebra I January 5{7, 2009 3 / 55 Motivation Why multilinear: \Classification of mathematical problems as linear and nonlinear is like classification of the Universe as bananas and non-bananas." Nonlinear | too general.
    [Show full text]
  • Tensor)Algebraic Frameworkfor Computer Graphics,Computer Vision, and Machine Learning
    AMULTILINEAR (TENSOR)ALGEBRAIC FRAMEWORK FOR COMPUTER GRAPHICS,COMPUTER VISION, AND MACHINE LEARNING by M. Alex O. Vasilescu A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Computer Science University of Toronto Copyright c 2009 by M. Alex O. Vasilescu A Multilinear (Tensor) Algebraic Framework for Computer Graphics, Computer Vision, and Machine Learning M. Alex O. Vasilescu Doctor of Philosophy Graduate Department of Computer Science University of Toronto 2009 Abstract This thesis introduces a multilinear algebraic framework for computer graphics, computer vision, and machine learning, particularly for the fundamental purposes of image synthesis, analysis, and recog- nition. Natural images result from the multifactor interaction between the imaging process, the scene illumination, and the scene geometry. We assert that a principled mathematical approach to disentan- gling and explicitly representing these causal factors, which are essential to image formation, is through numerical multilinear algebra, the algebra of higher-order tensors. Our new image modeling framework is based on (i) a multilinear generalization of principal compo- nents analysis (PCA), (ii) a novel multilinear generalization of independent components analysis (ICA), and (iii) a multilinear projection for use in recognition that maps images to the multiple causal factor spaces associated with their formation. Multilinear PCA employs a tensor extension of the conventional matrix singular value decomposition
    [Show full text]
  • Tensor, Exterior and Symmetric Algebras
    Tensor, Exterior and Symmetric Algebras Daniel Murfet May 16, 2006 Throughout this note R is a commutative ring, all modules are left R-modules. If we say a ring is noncommutative, we mean it is not necessarily commutative. Unless otherwise specified, all rings are noncommutative (except for R). If A is a ring then the center of A is the set of all x ∈ A with xy = yx for all y ∈ A. Contents 1 Definitions 1 2 The Tensor Algebra 5 3 The Exterior Algebra 6 3.1 Dimension of the Exterior Powers ............................ 11 3.2 Bilinear Forms ...................................... 14 3.3 Other Properties ..................................... 18 3.3.1 The determinant formula ............................ 18 4 The Symmetric Algebra 19 1 Definitions Definition 1. A R-algebra is a ring morphism φ : R −→ A where A is a ring and the image of φ is contained in the center of A. This is equivalent to A being an R-module and a ring, with r · (ab) = (r · a)b = a(r · b), via the identification of r · 1 and φ(r). A morphism of R-algebras is a ring morphism making the appropriate diagram commute, or equivalently a ring morphism which is also an R-module morphism. In this section RnAlg will denote the category of these R-algebras. We use RAlg to denote the category of commutative R-algebras. A graded ring is a ring A together with a set of subgroups Ad, d ≥ 0 such that A = ⊕d≥0Ad as an abelian group, and st ∈ Ad+e for all s ∈ Ad, t ∈ Ae.
    [Show full text]