Algebra-2: Groups and Rings
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The Universal Property of Inverse Semigroup Equivariant KK-Theory
KYUNGPOOK Math. J. 61(2021), 111-137 https://doi.org/10.5666/KMJ.2021.61.1.111 pISSN 1225-6951 eISSN 0454-8124 c Kyungpook Mathematical Journal The Universal Property of Inverse Semigroup Equivariant KK-theory Bernhard Burgstaller Departamento de Matematica, Universidade Federal de Santa Catarina, CEP 88.040-900 Florian´opolis-SC, Brasil e-mail : [email protected] Abstract. Higson proved that every homotopy invariant, stable and split exact func- tor from the category of C∗-algebras to an additive category factors through Kasparov's KK-theory. By adapting a group equivariant generalization of this result by Thomsen, we generalize Higson's result to the inverse semigroup and locally compact, not necessarily Hausdorff groupoid equivariant setting. 1. Introduction In [3], Cuntz noted that if F is a homotopy invariant, stable and split exact func- tor from the category of separable C∗-algebras to the category of abelian groups then Kasparov's KK-theory acts on F , that is, every element of KK(A; B) induces a natural map F (A) ! F (B). Higson [4], on the other hand, developed Cuntz' find- ings further and proved that every such functor F factorizes through the category K consisting of separable C∗-algebras as the object class and KK-theory together with the Kasparov product as the morphism class, that is, F is the composition F^ ◦ κ of a universal functor κ from the class of C∗-algebras to K and a functor F^ from K to abelian groups. In [12], Thomsen generalized Higson's findings to the group equivariant setting by replacing everywhere in the above statement algebras by equivariant algebras, ∗- homomorphisms by equivariant ∗-homomorphisms and KK-theory by equivariant KK-theory (the proof is however far from such a straightforward replacement). -
An Introduction to Inverse Semigroups and Their Connections to C*-Algebras
CARLETON UNIVERSITY SCHOOL OF MATHEMATICS AND STATISTICS HONOURS PROJECT TITLE: An Introduction to Inverse Semigroups and Their Connections To C*-Algebras AUTHOR: Kyle Cheaney SUPERVISOR: Dr. Charles Starling DATE: August 24, 2020 AN INTRODUCTION TO INVERSE SEMIGROUPS AND THEIR CONNECTIONS TO C∗-ALGEBRAS KYLE CHEANEY: 101012022 Abstract. This paper sets out to provide the reader with a basic introduction to inverse semigroups, as well as a proof of the Wagner-Preston Representation Theorem. It then focuses on finite directed graphs, specifically the graph inverse semigroup and how properties of the semigroup correspond to properties of C∗ algebras associated to such graphs. 1. Introduction The study of inverse semigroups was founded in the 1950's by three mathematicians: Ehresmann, Preston, and Wagner, as a way to algebraically describe pseudogroups of trans- formations. The purpose of this paper is to introduce the topic of inverse semigroups and some of the basic properties and theorems. We use these basics to relate the inverse semi- group formed by a directed graph to the corresponding graph C∗ algebra. The study of such C∗ algebras is still an active topic of research in analysis, thus providing a connection between abstract algebraic objects and complex analysis. In section 2, we outline some preliminary definitions and results which will be our basic tools for working with inverse semigroups. Section 3 provides a more detailed introduction to inverse semigroups and contains a proof of the Wagner-Preston representation theorem, a cornerstone in the study of inverse semigroups. In section 4, we present some detailed examples of inverse semigroups, including proving said sets are in fact inverse semigroups and discussing what their idempotents look like. -
On the Semisimplicity of Group Algebras
ON THE SEMISIMPLICITY OF GROUP ALGEBRAS ORLANDO E. VILLAMAYOR1 In this paper we find sufficient conditions for a group algebra over a commutative ring to be semisimple. In particular, the case in which the group is abelian is solved for fields of characteristic zero, and, in a more general case, for semisimple commutative rings which are uniquely divisible by every integer. Under similar restrictions on the ring of coefficients, it is proved the semisimplicity of group alge- bras when the group is not abelian but the factor group module its center is locally finite.2 In connection with this problem, we study homological properties of group algebras generalizing some results of M. Auslander [l, Theorems 6 and 9]. In fact, Lemmas 3 and 4 give new proofs of Aus- lander's Theorems 6 and 9 in the case C=(l). 1. Notations. A group G will be called a torsion group if every ele- ment of G has finite order, it will be called locally finite if every finitely generated subgroup is finite and it will be called free (or free abelian) if it is a direct sum of infinite cyclic groups. Direct sum and direct product are defined as in [3]. Given a set of rings Ri, their direct product will be denoted by J{Ri. If G is a group and R is a ring, the group algebra generated by G over R will be denoted by R(G). In a ring R, radical and semisimplicity are meant in the sense of Jacobson [5]. A ring is called regular in the sense of von Neumann [7]. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
A Review of Commutative Ring Theory Mathematics Undergraduate Seminar: Toric Varieties
A REVIEW OF COMMUTATIVE RING THEORY MATHEMATICS UNDERGRADUATE SEMINAR: TORIC VARIETIES ADRIANO FERNANDES Contents 1. Basic Definitions and Examples 1 2. Ideals and Quotient Rings 3 3. Properties and Types of Ideals 5 4. C-algebras 7 References 7 1. Basic Definitions and Examples In this first section, I define a ring and give some relevant examples of rings we have encountered before (and might have not thought of as abstract algebraic structures.) I will not cover many of the intermediate structures arising between rings and fields (e.g. integral domains, unique factorization domains, etc.) The interested reader is referred to Dummit and Foote. Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary opera- tions + and , respectively named addition and multiplication, satisfying · (R, +) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. a, b, c R, (a b) c = a (b c)) , • and the distributive8 law holds2 (i.e.· a,· b, c ·R, (·a + b) c = a c + b c, a (b + c)= • a b + a c.) 8 2 · · · · · · Moreover, the ring is commutative if multiplication is commutative. The ring has an identity, conventionally denoted 1, if there exists an element 1 R s.t. a R, 1 a = a 1=a. 2 8 2 · ·From now on, all rings considered will be commutative rings (after all, this is a review of commutative ring theory...) Since we will be talking substantially about the complex field C, let us recall the definition of such structure. Definition 1.2 (Fields). -
Abstract Algebra
Abstract Algebra Paul Melvin Bryn Mawr College Fall 2011 lecture notes loosely based on Dummit and Foote's text Abstract Algebra (3rd ed) Prerequisite: Linear Algebra (203) 1 Introduction Pure Mathematics Algebra Analysis Foundations (set theory/logic) G eometry & Topology What is Algebra? • Number systems N = f1; 2; 3;::: g \natural numbers" Z = f:::; −1; 0; 1; 2;::: g \integers" Q = ffractionsg \rational numbers" R = fdecimalsg = pts on the line \real numbers" p C = fa + bi j a; b 2 R; i = −1g = pts in the plane \complex nos" k polar form re iθ, where a = r cos θ; b = r sin θ a + bi b r θ a p Note N ⊂ Z ⊂ Q ⊂ R ⊂ C (all proper inclusions, e.g. 2 62 Q; exercise) There are many other important number systems inside C. 2 • Structure \binary operations" + and · associative, commutative, and distributive properties \identity elements" 0 and 1 for + and · resp. 2 solve equations, e.g. 1 ax + bx + c = 0 has two (complex) solutions i p −b ± b2 − 4ac x = 2a 2 2 2 2 x + y = z has infinitely many solutions, even in N (thei \Pythagorian triples": (3,4,5), (5,12,13), . ). n n n 3 x + y = z has no solutions x; y; z 2 N for any fixed n ≥ 3 (Fermat'si Last Theorem, proved in 1995 by Andrew Wiles; we'll give a proof for n = 3 at end of semester). • Abstract systems groups, rings, fields, vector spaces, modules, . A group is a set G with an associative binary operation ∗ which has an identity element e (x ∗ e = x = e ∗ x for all x 2 G) and inverses for each of its elements (8 x 2 G; 9 y 2 G such that x ∗ y = y ∗ x = e). -
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. -
Lie Groups and Lie Algebras
2 LIE GROUPS AND LIE ALGEBRAS 2.1 Lie groups The most general definition of a Lie group G is, that it is a differentiable manifold with group structure, such that the multiplication map G G G, (g,h) gh, and the inversion map G G, g g−1, are differentiable.× → But we shall7→ not need this concept in full generality.→ 7→ In order to avoid more elaborate differential geometry, we will restrict attention to matrix groups. Consider the set of all invertible n n matrices with entries in F, where F stands either for the real (R) or complex× (C) numbers. It is easily verified to form a group which we denote by GL(n, F), called the general linear group in n dimensions over the number field F. The space of all n n matrices, including non- invertible ones, with entries in F is denoted by M(n,×F). It is an n2-dimensional 2 vector space over F, isomorphic to Fn . The determinant is a continuous function det : M(n, F) F and GL(n, F) = det−1(F 0 ), since a matrix is invertible → −{ } 2 iff its determinant is non zero. Hence GL(n, F) is an open subset of Fn . Group multiplication and inversion are just given by the corresponding familiar matrix 2 2 2 2 2 operations, which are differentiable functions Fn Fn Fn and Fn Fn respectively. × → → 2.1.1 Examples of Lie groups GL(n, F) is our main example of a (matrix) Lie group. Any Lie group that we encounter will be a subgroups of some GL(n, F). -
A Natural Transformation of the Spec Functor
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 10, 69-91 (1968) A Natural Transformation of the Spec Functor IAN G. CONNELL McGill University, Montreal, Canada Communicated by P. Cohn Received November 28, 1967 The basic functor underlying the Grothendieck algebraic geometry is Spec which assigns a ringed space to each ring1 A; we denote the topological space by spec A and the sheaf of rings by Spec A. We shall define a functor which is both a generalization and a natural transformation of Spec, in a sense made precise below. Our setting is the category Proj, of commutative K-algebras, where k is a fixed ring, and specializations over K which are defined below. A k-algebra is, strictly speaking, a ring homomorphism pA : K -+ A (pa is called the representation or structural homomorphism) and a K-algebra homomorphism is a commutative triangle k of ring homomorphisms. As is customary we simplify this to “A is a K-algebra” and “4 : A -+ B is a K-algebra homomorphism”, when there can be no ambiguity about the p’s. When we refer to K as a K-algebra we mean of course the identity representation. 1. PRIMES AND PLACES Recall the definition of a place $ on the field A with values in the field B (cf. [6], p. 3). It is a ring homomorphism 4 : A, -+ B defined on a subring A, of A with the property that for all X, y E A, xy E A, , x 6 A, =Py E Ker 4. -
Math 250A: Groups, Rings, and Fields. H. W. Lenstra Jr. 1. Prerequisites
Math 250A: Groups, rings, and fields. H. W. Lenstra jr. 1. Prerequisites This section consists of an enumeration of terms from elementary set theory and algebra. You are supposed to be familiar with their definitions and basic properties. Set theory. Sets, subsets, the empty set , operations on sets (union, intersection, ; product), maps, composition of maps, injective maps, surjective maps, bijective maps, the identity map 1X of a set X, inverses of maps. Relations, equivalence relations, equivalence classes, partial and total orderings, the cardinality #X of a set X. The principle of math- ematical induction. Zorn's lemma will be assumed in a number of exercises. Later in the course the terminology and a few basic results from point set topology may come in useful. Group theory. Groups, multiplicative and additive notation, the unit element 1 (or the zero element 0), abelian groups, cyclic groups, the order of a group or of an element, Fermat's little theorem, products of groups, subgroups, generators for subgroups, left cosets aH, right cosets, the coset spaces G=H and H G, the index (G : H), the theorem of n Lagrange, group homomorphisms, isomorphisms, automorphisms, normal subgroups, the factor group G=N and the canonical map G G=N, homomorphism theorems, the Jordan- ! H¨older theorem (see Exercise 1.4), the commutator subgroup [G; G], the center Z(G) (see Exercise 1.12), the group Aut G of automorphisms of G, inner automorphisms. Examples of groups: the group Sym X of permutations of a set X, the symmetric group S = Sym 1; 2; : : : ; n , cycles of permutations, even and odd permutations, the alternating n f g group A , the dihedral group D = (1 2 : : : n); (1 n 1)(2 n 2) : : : , the Klein four group n n h − − i V , the quaternion group Q = 1; i; j; ij (with ii = jj = 1, ji = ij) of order 4 8 { g − − 8, additive groups of rings, the group Gl(n; R) of invertible n n-matrices over a ring R. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
Notes D1: Group Rings
2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 1. The group ring k[G] The main idea is that representations of a group G over a field k are “the same” as modules over the group ring k[G]. First I defined both terms. 1.1. Representations of groups. Definition 1.1. A representation of a group G over a field k is defined to be a group homomorphism ρ : G Aut (V ) → k where V is a vector space over k. Here Autk(V ) is the group of k-linear automorphisms of V . This also written as GLk(V ). This is the group of units of the ring Endk(V )= Homk(V, V ) which, as I explained before, is a ring with addition defined pointwise and multiplication given by composition. If dimk(V )=d d then Autk(V ) ∼= Autk(k )=GLd(k) which can also be described as the group of units of the ring Matd(k) or as: GL (k)= A Mat (k) det(A) =0 d { ∈ d | $ } d = dimk(V ) is called the dimension of the representation ρ. 1.1.1. examples. Example 1.2. The first example I gave was the trivial representation. This is usually defined to be the one dimensional representation V = k with trivial action of the group G (which can be arbitrary). Trivial action means that ρ(σ) = 1 = id for all σ G. V ∈ In the next example, I pointed out that the group G needs to be written multiplicatively no matter what. Example 1.3. Let G = Z/3.