DOCSLIB.ORG

Determination of the Differentiably Simple Rings with a Minimal Ideal Author(S): Richard E

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Annals of Mathematics Determination of the Differentiably Simple Rings with a Minimal Ideal Author(s): Richard E. Block Source: The Annals of Mathematics, Second Series, Vol. 90, No. 3 (Nov., 1969), pp. 433-459 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970745 Accessed: 07/11/2010 19:03 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=annals. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics. http://www.jstor.org Determinationof the differentiablysimple ringswith a minimalideal* By RICHARD E. BLOCK 1. In.troduc tionl. A centralresult in ring theoryis the Wedderburn-Artintheorem on sim- ple rings: a simpleassociative ring with DCC on left ideals is a total matrix ringA., over a divisionring A\. In this paper we consideran analogue of this theoremin which ideals are replacedby differentialideals, these being ideals invariantunder all derivationsof the ring, and left ideals are replaced by ideals. The main result is a completedetermination of the differentiably simplerings with a minimalideal, in termsof the simple rings. This result (the precisestatement will be givenshortly) is new even forfinite-dimensional associative algebras with a unit. However, the result holds also for com- pletelyarbitrary rings, not necessarilyassociative and not necessarilyhaving a unit element(just differentiablysimple with a minimalideal). In the case of Lie algebras the theoremproves a thirty-year-oldconjecture of Zassenhaus. In fact the resultleads to the solutionof importantproblems on two verydif- ferentclasses of finite-dimensionalnon-associative algebras, namely,semi- simple Lie algebras (of characteristicp) and simple flexible(but not anti- commutative)power-associative algebras. We now give somedefinitions and notation. If A is a ringand if D is a set of derivationsof A (additive mappings d of A into A such that d(ab) = (da)b + a(db) forall a, b in A) thenby a D-ideal of A is meantan ideal of A invariantunder D. The ring is called D-simple (d-simpleif D consistsof a single derivationd) if A2 # 0 and if A has no proper D-ideals. Also A is called differentiablysimple if it is D-simplefor some set of derivationsD of A, and hence forthe set of all derivationsof A. The same definitionsare used foralgebras over a ring K, the derivationsthen being assumed to be K- linear. Note that when we use the wordring or algebra we do not in general assume, unless stated, that the ring or algebra is associative or has a unit element; however,when we speak of an algebra over a ring,say over K, we * This research was supported by the National Science Foundation under grants GP 5949 through the University of Illinois, GP 6558 throughYale University,and GP 8713 throughthe Universityof California,Riverside. 434 RICHARD E. BLOCK always assume that K is associative with a unit elementacting unitallyon the algebra. Jacobsonnoted (at least in a special case, see [16]) the followingclass of examples of differentiablysimple rings A which are not simple: A is the group ring SG whereS is a simplering of primecharacteristic p and G # 1 is a finiteelementary abelian p-group(so that G is a direct productof say n copies (n > 1) of the cyclicgroup of orderp). If S is an algebra overK then SG is also an algebra over K. Since the ring or algebra SG depends (up to isomorphism)only on S and n, we introducethe notationS[,,] for it. We only use this notation when S is simple of prime characteristic. We also let S[,] denote S itself. If S is an algebra over K (so K= Z or ZP in the ring case) then S[n] may also be writtenas S($? Bf,p(K) where Bn,P(K) denotesthe (commutativeassociative with unit) truncatedpolynomial algebra *I,. X]/(XP, *. *, XP) (p the characteristic of S). Often one is inter- ested in the case in which K is a fieldof characteristicp, when Bn,p(K) is usually just writtenB,(K). Note that S[n] is associative or Lie etc. accord- ing as S is. We now state the most importantresult of the paper. MAIN THEOREM. If A is a differentiably simple ring (or K-algebra) with a minimal ideal, theneither A is simple or thereis a simple ring (or K-algebra) S of prime characteristic and a positive integer n such that A = S[n]. Converselyit is easy to see (and is a special case of results given below) that each S[n]is differentiablysimple and has a minimalideal, in fact a unique minimalideal. The assumptionthat A has a minimalideal cannotbe removed, as is shownfor example by polynomialand powerseries ringsover a fieldof characteristic0. Posner [11] notedthat any differentiablysimple ring may be regardedas an algebra over a field(its differentialcentroid) and so has a characteristic. Three special cases of the theoremabove were known. At characteristic0, wherethe situationis comparativelyuncomplicated, these were due to Posner [11] (the associative case) and Sagle and Winter [15] (the case of finite-di- mensionalalgebras). In the much more complicatedcharacteristic p case, Harper [6] proved the result for finite-dimensionalcommutative associative algebrasover an algebraicallyclosed field F (the resultthen being A - RJF)), whichproved a conjectureof Albert [3]. In the case of finite-dimensional Lie algebras of characteristicp the problemwas also studied by Zassenhaus [17] and Seligman[16]. Zassenhaus [17, p. 80] conjecturedthe result in this DIFFERENTIABLY SIMPLE RINGS 435 case in what we shall show in ?6 is an equivalentform. The above resulton the non-simpledifferentiably simple rings is as men- tioned analogous to the Wedderburn-Artintheorem. One aspect of that analogy is that the role of the divisionrings in that theoremis played here by simplerings, and in the associative case divisionrings are preciselyanalo- gous to simple rings in that they may be characterizedas the non-trivial ringswith no left ideals. Most of the proofof the Main Theorem(all of it in the characteristic0 or finite-dimensionalassociative cases) is valid fora D- simplering A (with minimalideal) when each operatord in D satisfiesa re- quirementweaker than that of being a derivation,namely d is additiveand [d, t] e T(A) for each t in T(A) where T(A) denotesthe ringgenerated by all rightand left multiplicationsby elementsof A. We call such an operatord a quasi-derivation. The proofof the Main Theoremoccupies ?2-?6. The followingis an out- line of this proof. A key fact proved in ?2 is that A, as a module for its multiplicationalgebra, has a compositionseries with isomorphic factors. Also A has a unique maximalideal N. In ?3 it is shownthat the centroidC of A is also differentiablysimple with a minimalideal, and that if C containsa sub- fieldE such that A/N is centralover E and if A (as an E algebra) containsa subalgebraS -A/N with S + N = A then A S 0 EC; moreoverit is es- sentiallyshown that such an S exists if A is d-simplefor some derivationd. A proofof the commutativeassociative case is given in ?4, thus determining C. In ?5 the d used to constructS is shownto exist by provingTheorem 5.2: if a ring A witha minimal ideal is D-simple whereD is a set of (quasi)- derivationsclosed under addition, commutationand left multiplication by all elementsof the centroidthen there is a d in D such that A is d-simple. (As a veryspecial case, any differentiablysimple ring A with minimalideal is d-simplefor some derivationd; when A = Bn(F) this gives a result of Albert [3]). In ?6 the requiredfield E is constructed,to completethe proof. In ?7 we shall determinethe derivationalgebra der S[i] of S[o] in terms of der S, and give a conditionon a set D of derivationsof S[,f]for S[,,] to be D-simple. In ?8 we shall considerdifferentiably semisimple rings and give an ana- logue of the followingWedderburn-Artin theorem: a semisimpleassociative ringwith DCC on left ideals is a directsum of simple rings. If D is a set of (quasi)-derivationsof a ring A and if I is an ideal of A let ID denote the (unique) largest D-ideal of A containedin I. We shall provein Theorem8.2 that if A has DCC on ideals and if A/I is a direct sum of simple rings then A/IL)is a direct sum of rings which are D-simple (or more preciselysimple 436 RICHARD E. BLOCK with respectto the set of (quasi)-derivationsinduced on them by D). If A is associative and R is the (Jacobson)radical of A, then we say that A is D- semisimpleif RD = 0. As a corollary,if A is associative and D-semisimmple with DCC on ideals and if AIR has DCC on left ideals, then A is a direct sum of D-simple rings (and so is known,in termsof divisionrings).
Recommended publications
  • Single Elements

    Single Elements

    Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 47 (2006), No. 1, 275-288. Single Elements B. J. Gardner Gordon Mason Department of Mathematics, University of Tasmania Hobart, Tasmania, Australia Department of Mathematics & Statistics, University of New Brunswick Fredericton, N.B., Canada Abstract. In this paper we consider single elements in rings and near- rings. If R is a (near)ring, x ∈ R is called single if axb = 0 ⇒ ax = 0 or xb = 0. In seeking rings in which each element is single, we are led to consider 0-simple rings, a class which lies between division rings and simple rings. MSC 2000: 16U99 (primary), 16Y30 (secondary) Introduction If R is a (not necessarily unital) ring, an element s is called single if whenever asb = 0 then as = 0 or sb = 0. The definition was first given by J. Erdos [2] who used it to obtain results in the representation theory of normed algebras. More recently the idea has been applied by Longstaff and Panaia to certain matrix algebras (see [9] and its bibliography) and they suggest it might be worthy of further study in other contexts. In seeking rings in which every element is single we are led to consider 0-simple rings, a class which lies between division rings and simple rings. In the final section we examine the situation in nearrings and obtain information about minimal N-subgroups of some centralizer nearrings. 1. Single elements in rings We begin with a slightly more general definition. If I is a one-sided ideal in a ring R an element x ∈ R will be called I-single if axb ∈ I ⇒ ax ∈ I or xb ∈ I.
  • Modules and Lie Semialgebras Over Semirings with a Negation Map 3

    Modules and Lie Semialgebras Over Semirings with a Negation Map 3

    MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP GUY BLACHAR Abstract. In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan’s criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem. Contents Page 0. Introduction 2 0.1. Semirings with a Negation Map 2 0.2. Modules Over Semirings with a Negation Map 3 0.3. Supertropical Algebras 4 0.4. Exploded Layered Tropical Algebras 4 1. Modules over Semirings with a Negation Map 5 1.1. The Surpassing Relation for Modules 6 1.2. Basic Definitions for Modules 7 1.3. -morphisms 9 1.4. Lifting a Module Over a Semiring with a Negation Map 10 1.5. Linearly Independent Sets 13 1.6. d-bases and s-bases 14 1.7. Free Modules over Semirings with a Negation Map 18 2.
  • LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie

    LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie

    LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1.
  • Right Ideals of a Ring and Sublanguages of Science

    Right Ideals of a Ring and Sublanguages of Science

    RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance.
  • Quaternion Algebras and Modular Forms

    Quaternion Algebras and Modular Forms

    QUATERNION ALGEBRAS AND MODULAR FORMS JIM STANKEWICZ We wish to know about generalizations of modular forms using quater- nion algebras. We begin with preliminaries as follows. 1. A few preliminaries on quaternion algebras As we must to make a correct statement in full generality, we begin with a profoundly unhelpful definition. Definition 1. A quaternion algebra over a field k is a 4 dimensional vector space over k with a multiplication action which turns it into a central simple algebra Four dimensional vector spaces should be somewhat familiar, but what of the rest? Let's start with the basics. Definition 2. An algebra B over a ring R is an R-module with an associative multiplication law(hence a ring). The most commonly used examples of such rings in arithmetic geom- etry are affine polynomial rings R[x1; : : : ; xn]=I where R is a commu- tative ring and I an ideal. We can have many more examples though. Example 1. If R is a ring (possibly non-commutative), n 2 Z≥1 then the ring of n by n matrices over R(henceforth, Mn(R)) form an R- algebra. Definition 3. A simple ring is a ring whose only 2-sided ideals are itself and (0) Equivalently, a ring B is simple if for any ring R and any nonzero ring homomorphism φ : B ! R is injective. We show here that if R = k and B = Mn(k) then B is simple. Suppose I is a 2-sided ideal of B. In particular, it is a right ideal, so BI = I.
  • CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V

    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.
  • On Various Types of Ideals of Gamma Rings and the Corresponding Operator Rings

    On Various Types of Ideals of Gamma Rings and the Corresponding Operator Rings

    Ranu Paul Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 8, (Part -4) August 2015, pp.95-98 RESEARCH ARTICLE OPEN ACCESS On Various Types of Ideals of Gamma Rings and the Corresponding Operator Rings Ranu Paul Department of Mathematics Pandu College, Guwahati 781012 Assam State, India Abstract The prime objective of this paper is to prove some deep results on various types of Gamma ideals. The characteristics of various types of Gamma ideals viz. prime/maximal/minimal/nilpotent/primary/semi-prime ideals of a Gamma-ring are shown to be maintained in the corresponding right (left) operator rings of the Gamma-rings. The converse problems are also investigated with some good outcomes. Further it is shown that the projective product of two Gamma-rings cannot be simple. AMS Mathematics subject classification: Primary 16A21, 16A48, 16A78 Keywords: Ideals/Simple Gamma-ring/Projective Product of Gamma-rings I. Introduction left) ideal if the only right (or left) ideal of 푋 Ideals are the backbone of the Gamma-ring contained in 퐼 are 0 and 퐼 itself. theory. Nobusawa developed the notion of a Gamma- ring which is more general than a ring [3]. He Definition 2.5: A nonzero ideal 퐼 of a gamma ring obtained the analogue of the Wedderburn theorem for (푋, ) such that 퐼 ≠ 푋 is said to be maximal ideal, simple Gamma-ring with minimum condition on one- if there exists no proper ideal of 푋 containing 퐼. sided ideals. Barnes [6] weakened slightly the defining conditions for a Gamma-ring, introduced the Definition 2.6: An ideal 퐼 of a gamma ring (푋, ) is notion of prime ideals, primary ideals and radical for said to be primary if it satisfies, a Gamma-ring, and obtained analogues of the 푎훾푏 ⊆ 퐼, 푎 ⊈ 퐼 => 푏 ⊆ 퐽 ∀ 푎, 푏 ∈ 푋 푎푛푑 훾 ∈ .
  • Introuduction to Representation Theory of Lie Algebras

    Introuduction to Representation Theory of Lie Algebras

    Introduction to Representation Theory of Lie Algebras Tom Gannon May 2020 These are the notes for the summer 2020 mini course on the representation theory of Lie algebras. We'll first define Lie groups, and then discuss why the study of representations of simply connected Lie groups reduces to studying representations of their Lie algebras (obtained as the tangent spaces of the groups at the identity). We'll then discuss a very important class of Lie algebras, called semisimple Lie algebras, and we'll examine the repre- sentation theory of two of the most basic Lie algebras: sl2 and sl3. Using these examples, we will develop the vocabulary needed to classify representations of all semisimple Lie algebras! Please email me any typos you find in these notes! Thanks to Arun Debray, Joakim Færgeman, and Aaron Mazel-Gee for doing this. Also{thanks to Saad Slaoui and Max Riestenberg for agreeing to be teaching assistants for this course and for many, many helpful edits. Contents 1 From Lie Groups to Lie Algebras 2 1.1 Lie Groups and Their Representations . .2 1.2 Exercises . .4 1.3 Bonus Exercises . .4 2 Examples and Semisimple Lie Algebras 5 2.1 The Bracket Structure on Lie Algebras . .5 2.2 Ideals and Simplicity of Lie Algebras . .6 2.3 Exercises . .7 2.4 Bonus Exercises . .7 3 Representation Theory of sl2 8 3.1 Diagonalizability . .8 3.2 Classification of the Irreducible Representations of sl2 .............8 3.3 Bonus Exercises . 10 4 Representation Theory of sl3 11 4.1 The Generalization of Eigenvalues .
  • Ring (Mathematics) 1 Ring (Mathematics)

    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.
  • Matrix Lie Groups and Their Lie Algebras

    Matrix Lie Groups and Their Lie Algebras

    Matrix Lie groups and their Lie algebras Alen Alexanderian∗ Abstract We discuss matrix Lie groups and their corresponding Lie algebras. Some common examples are provided for purpose of illustration. 1 Introduction The goal of these brief note is to provide a quick introduction to matrix Lie groups which are a special class of abstract Lie groups. Study of matrix Lie groups is a fruitful endeavor which allows one an entry to theory of Lie groups without requiring knowl- edge of differential topology. After all, most interesting Lie groups turn out to be matrix groups anyway. An abstract Lie group is defined to be a group which is also a smooth manifold, where the group operations of multiplication and inversion are also smooth. We provide a much simple definition for a matrix Lie group in Section 4. Showing that a matrix Lie group is in fact a Lie group is discussed in standard texts such as [2]. We also discuss Lie algebras [1], and the computation of the Lie algebra of a Lie group in Section 5. We will compute the Lie algebras of several well known Lie groups in that section for the purpose of illustration. 2 Notation Let V be a vector space. We denote by gl(V) the space of all linear transformations on V. If V is a finite-dimensional vector space we may put an arbitrary basis on V and identify elements of gl(V) with their matrix representation. The following define various classes of matrices on Rn: ∗The University of Texas at Austin, USA. E-mail: [email protected] Last revised: July 12, 2013 Matrix Lie groups gl(n) : the space of n
  • What Does a Lie Algebra Know About a Lie Group?

    What Does a Lie Algebra Know About a Lie Group?

    WHAT DOES A LIE ALGEBRA KNOW ABOUT A LIE GROUP? HOLLY MANDEL Abstract. We define Lie groups and Lie algebras and show how invariant vector fields on a Lie group form a Lie algebra. We prove that this corre- spondence respects natural maps and discuss conditions under which it is a bijection. Finally, we introduce the exponential map and use it to write the Lie group operation as a function on its Lie algebra. Contents 1. Introduction 1 2. Lie Groups and Lie Algebras 2 3. Invariant Vector Fields 3 4. Induced Homomorphisms 5 5. A General Exponential Map 8 6. The Campbell-Baker-Hausdorff Formula 9 Acknowledgments 10 References 10 1. Introduction Lie groups provide a mathematical description of many naturally-occuring sym- metries. Though they take a variety of shapes, Lie groups are closely linked to linear objects called Lie algebras. In fact, there is a direct correspondence between these two concepts: simply-connected Lie groups are isomorphic exactly when their Lie algebras are isomorphic, and every finite-dimensional real or complex Lie algebra occurs as the Lie algebra of a simply-connected Lie group. To put it another way, a simply-connected Lie group is completely characterized by the small collection n2(n−1) of scalars that determine its Lie bracket, no more than 2 numbers for an n-dimensional Lie group. In this paper, we introduce the basic Lie group-Lie algebra correspondence. We first define the concepts of a Lie group and a Lie algebra and demonstrate how a certain set of functions on a Lie group has a natural Lie algebra structure.
  • Clifford Algebras and Lie Groups

    Clifford Algebras and Lie Groups

    Clifford algebras and Lie groups Eckhard Meinrenken Lecture Notes, University of Toronto, Fall 2009. (Revised version of lecture notes from 2005) CHAPTER 1 Symmetric bilinear forms Throughout, K will denote a field of characteristic = 2. We are mainly interested in the cases K = R or C, and sometimes specialize6 to those two cases. 1. Quadratic vector spaces Suppose V is a finite-dimensional vector space over K. For any bilinear form B : V V K, define a linear map × → ♭ B : V V ∗, v B(v, ). → 7→ · The bilinear form B is called symmetric if it satisfies B(v1, v2)= B(v2, v1) ♭ ♭ for all v1, v2 V . Since dim V < this is equivalent to (B )∗ = B . The symmetric∈ bilinear form B is uniquely∞ determined by the associated quadratic form, QB(v)= B(v, v) by the polarization identity, (1) B(v, w)= 1 Q (v + w) Q (v) Q (w) . 2 B − B − B The kernel (also called radical ) of B is the subspace ker(B)= v V B(v, v ) = 0 for all v V , { ∈ | 1 1 ∈ } i.e. the kernel of the linear map B♭. The bilinear form B is called non- degenerate if ker(B) = 0, i.e. if and only if B♭ is an isomorphism. A vector space V together with a non-degenerate symmetric bilinear form B will be referred to as a quadratic vector space. Assume for the rest of this chapter that (V,B) is a quadratic vector space. Definition 1.1. A vector v V is called isotropic if B(v, v) = 0, and non-isotropic if B(v, v) = 0.