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Exercises and Solutions in Groups Rings and Fields
EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuo˘glu Middle East Technical University [email protected] Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . v 1. SETS, INTEGERS, FUNCTIONS . 1 2. GROUPS . 4 3. RINGS . .55 4. FIELDS . 77 5. INDEX . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract al- gebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank B¨ulent B¨uy¨ukbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. Ismail_ S¸. G¨ulo˘glufor checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and G¨uldaneG¨um¨u¸sfor their typing the manuscript in LATEX. Mahmut Kuzucuo˘glu I would like to thank our graduate students Tu˘gbaAslan, B¨u¸sra C¸ınar, Fuat Erdem and Irfan_ Kadık¨oyl¨ufor reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuo˘glu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2n distinct subsets. -
Arxiv:2008.02889V1 [Math.QA]
NONCOMMUTATIVE NETWORKS ON A CYLINDER S. ARTHAMONOV, N. OVENHOUSE, AND M. SHAPIRO Abstract. In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder Σ, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra kπ1(Σ,p) of the fundamental group of a surface based at p ∈ ∂Σ. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space C♮, which is a k-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of [Ove20] that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on C♮. 1. Introduction The current manuscript is obtained as a continuation of papers [Ove20, FK09, DF15, BR11] where the authors develop noncommutative generalizations of discrete completely integrable dynamical systems and [BR18] where a large class of noncommutative cluster algebras was constructed. Cluster algebras were introduced in [FZ02] by S.Fomin and A.Zelevisnky in an effort to describe the (dual) canonical basis of universal enveloping algebra U(b), where b is a Borel subalgebra of a simple complex Lie algebra g. Cluster algebras are commutative rings of a special type, equipped with a distinguished set of generators (cluster variables) subdivided into overlapping subsets (clusters) of the same cardinality subject to certain polynomial relations (cluster transformations). -
Using March Madness in the First Linear Algebra Course
Using March Madness in the first Linear Algebra course Steve Hilbert Ithaca College [email protected] Background • National meetings • Tim Chartier • 1 hr talk and special session on rankings • Try something new Why use this application? • This is an example that many students are aware of and some are interested in. • Interests a different subgroup of the class than usual applications • Interests other students (the class can talk about this with their non math friends) • A problem that students have “intuition” about that can be translated into Mathematical ideas • Outside grading system and enforcer of deadlines (Brackets “lock” at set time.) How it fits into Linear Algebra • Lots of “examples” of ranking in linear algebra texts but not many are realistic to students. • This was a good way to introduce and work with matrix algebra. • Using matrix algebra you can easily scale up to work with relatively large systems. Filling out your bracket • You have to pick a winner for each game • You can do this any way you want • Some people use their “ knowledge” • I know Duke is better than Florida, or Syracuse lost a lot of games at the end of the season so they will probably lose early in the tournament • Some people pick their favorite schools, others like the mascots, the uniforms, the team tattoos… Why rank teams? • If two teams are going to play a game ,the team with the higher rank (#1 is higher than #2) should win. • If there are a limited number of openings in a tournament, teams with higher rankings should be chosen over teams with lower rankings. -
The Cardinality of the Center of a Pi Ring
Canad. Math. Bull. Vol. 41 (1), 1998 pp. 81±85 THE CARDINALITY OF THE CENTER OF A PI RING CHARLES LANSKI ABSTRACT. The main result shows that if R is a semiprime ring satisfying a poly- nomial identity, and if Z(R) is the center of R, then card R Ä 2cardZ(R). Examples show that this bound can be achieved, and that the inequality fails to hold for rings which are not semiprime. The purpose of this note is to compare the cardinality of a ring satisfying a polynomial identity (a PI ring) with the cardinality of its center. Before proceeding, we recall the def- inition of a central identity, a notion crucial for us, and a basic result about polynomial f g≥ f g identities. Let C be a commutative ring with 1, F X Cn x1,...,xn the free algebra f g ≥ 2 f gj over C in noncommuting indeterminateso xi ,andsetG f(x1,...,xn) F X some coef®cient of f is a unit in C .IfRis an algebra over C,thenf(x1,...,xn)2Gis a polyno- 2 ≥ mial identity (PI) for RPif for all ri R, f (r1,...,rn) 0. The standard identity of degree sgõ n is Sn(x1,...,xn) ≥ õ(1) xõ(1) ÐÐÐxõ(n) where õ ranges over the symmetric group on n letters. The Amitsur-Levitzki theorem is an important result about Sn and shows that Mk(C) satis®es Sn exactly for n ½ 2k [5; Lemma 2, p. 18 and Theorem, p. 21]. Call f 2 G a central identity for R if f (r1,...,rn) 2Z(R), the center of R,forallri 2R, but f is not a polynomial identity. -
SCHUR-WEYL DUALITY Contents Introduction 1 1. Representation
SCHUR-WEYL DUALITY JAMES STEVENS Contents Introduction 1 1. Representation Theory of Finite Groups 2 1.1. Preliminaries 2 1.2. Group Algebra 4 1.3. Character Theory 5 2. Irreducible Representations of the Symmetric Group 8 2.1. Specht Modules 8 2.2. Dimension Formulas 11 2.3. The RSK-Correspondence 12 3. Schur-Weyl Duality 13 3.1. Representations of Lie Groups and Lie Algebras 13 3.2. Schur-Weyl Duality for GL(V ) 15 3.3. Schur Functors and Algebraic Representations 16 3.4. Other Cases of Schur-Weyl Duality 17 Appendix A. Semisimple Algebras and Double Centralizer Theorem 19 Acknowledgments 20 References 21 Introduction. In this paper, we build up to one of the remarkable results in representation theory called Schur-Weyl Duality. It connects the irreducible rep- resentations of the symmetric group to irreducible algebraic representations of the general linear group of a complex vector space. We do so in three sections: (1) In Section 1, we develop some of the general theory of representations of finite groups. In particular, we have a subsection on character theory. We will see that the simple notion of a character has tremendous consequences that would be very difficult to show otherwise. Also, we introduce the group algebra which will be vital in Section 2. (2) In Section 2, we narrow our focus down to irreducible representations of the symmetric group. We will show that the irreducible representations of Sn up to isomorphism are in bijection with partitions of n via a construc- tion through certain elements of the group algebra. -
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. -
The Notion Of" Unimaginable Numbers" in Computational Number Theory
Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers. -
Interval Notation (Mixed Parentheses and Brackets) for Interval Notation Using Parentheses on Both Sides Or Brackets on Both Sides, You Can Use the Basic View
Canvas Equation Editor Tips: Advanced View With the Advanced View of the Canvas Equation Editor, you can input more complicated mathematical text using LaTeX, like matrices, interval notation, and piecewise functions. In case you don’t have a lot of LaTeX experience, here are some tips for how to write LaTex for some commonly used mathematics. More tips can be found in the Basic View Tips PDF. Interval Notation (mixed parentheses and brackets) For interval notation using parentheses on both sides or brackets on both sides, you can use the basic view. Just type ( to get ( ) or [ to get [ ] in the editor. Left parenthesis \left( infinity symbol \infty Left bracket \left[ negative infinity -\infty Right parenthesis \right) square root \sqrt{ } Right bracket \right] fraction \frac{ }{ } Below are several examples that should help you construct the LaTeX for the interval notation you need. Display LaTeX in Advanced View \left(x,y\right] \left[x,y\right) \left[-2,\infty\right) \left(-\infty,4\right] \left(\sqrt{5} ,\frac{3}{4}\right] © Canvas 2021 | guides.canvaslms.com | updated 2017-12-09 Page 1 Canvas Equation Editor Tips: Advanced View Piecewise Functions You can use the Advanced View of the Canvas Equation Editor to write piecewise functions. To do this, you will need to write the appropriate LaTeX to express the mathematical text (see example below): This kind of LaTeX expression is like an onion - there are many layers. The outer layer declares the function and a resizing brace: The next layer in builds an array to hold the function definitions and conditions: The innermost layer defines the functions and conditions. -
Redundant Cloud Services Skyrocketing! How to Provision Minecraft Server on Metacloud
Cisco Service Provider Cloud Josip Zimet CCIE 5688 Cisco My Favorite Example of Digital Transformations started with data centers …. Example of Digital Transformations started with data centers …. Paris Dubai Dubrovnik https://developer.cisco.com/site/flare/ SIM Card Identity for a Phone + SIM Card Identity for a Phone HSRP x.x.x.1 + STP/802.1Q/FP IS-IS/BGP/VXLAN Anycast GW x.x.x.1 Physical, virtual, Container SIM Card Identity for a Phone + Multitenant multivendor across bare metal, virtual and container private and public cloud Cloud Center VMs=house Apartments=containers Nova/cinder/Neutron NSX/ACI/Contiv EC2/S3/EBS/VPC/Sec Groups Broad Multi-Vendor Infrastructure Support UCS Director Converged VM L4-L7 Compute Network Storage vASA, Nexus CSR1000v MDS * * * * * * * * * Partner provided roleback https://www.youtube.com/watch?v=hz7zwd98rn4 No Web-1 No Web-2 No App-1 No DB-1 No DB-2 No DB-3 No App-2 No DB-3 Value of Sec ? st 0.36 Seconds nd 1 Place ($881,000) separates 2 Place 1st and 2nd place • $2,447 Per Millisecond • $1.6 more dollars awarded to • $719,000 Million 1st Place Value of Sec ? Financial Media Transport Retail Airline Brokerage Home Shopping Pay per View Reservations operations $1,883/min $2,500/min $1,483/min $107,500/min Credit card/Sales Authorizations Teleticket Package Catalogue Sales $43,333/min Sales Shipping $1,500/min $1,150/min $466/min ATM Fees $241/min Classification Availability Annual Down Time Continuous processing 100% 0 min/year Fault Tolerant 99-999% 5 min/Year Fault Resilient 99.99% 53 min/year High Availability -
Matrices Lie: an Introduction to Matrix Lie Groups and Matrix Lie Algebras
Matrices Lie: An introduction to matrix Lie groups and matrix Lie algebras By Max Lloyd A Journal submitted in partial fulfillment of the requirements for graduation in Mathematics. Abstract: This paper is an introduction to Lie theory and matrix Lie groups. In working with familiar transformations on real, complex and quaternion vector spaces this paper will define many well studied matrix Lie groups and their associated Lie algebras. In doing so it will introduce the types of vectors being transformed, types of transformations, what groups of these transformations look like, tangent spaces of specific groups and the structure of their Lie algebras. Whitman College 2015 1 Contents 1 Acknowledgments 3 2 Introduction 3 3 Types of Numbers and Their Representations 3 3.1 Real (R)................................4 3.2 Complex (C).............................4 3.3 Quaternion (H)............................5 4 Transformations and General Geometric Groups 8 4.1 Linear Transformations . .8 4.2 Geometric Matrix Groups . .9 4.3 Defining SO(2)............................9 5 Conditions for Matrix Elements of General Geometric Groups 11 5.1 SO(n) and O(n)........................... 11 5.2 U(n) and SU(n)........................... 14 5.3 Sp(n)................................. 16 6 Tangent Spaces and Lie Algebras 18 6.1 Introductions . 18 6.1.1 Tangent Space of SO(2) . 18 6.1.2 Formal Definition of the Tangent Space . 18 6.1.3 Tangent space of Sp(1) and introduction to Lie Algebras . 19 6.2 Tangent Vectors of O(n), U(n) and Sp(n)............. 21 6.3 Tangent Space and Lie algebra of SO(n).............. 22 6.4 Tangent Space and Lie algebras of U(n), SU(n) and Sp(n).. -
Math 547–Review Problems–Exam 1 Note That Problem 8 Had the Word ‘Commutative’ Inserted to Make It a Little Nicer
Math 547–Review Problems–Exam 1 Note that problem 8 had the word ‘commutative’ inserted to make it a little nicer. Be able to do problems such as those below and the problems assigned on problem sets or the following problems from your textbook. Textbook Problems: Page 110–111: 1, 2, 3, 4, 7, 9, 13 14(c), 15, 18, 20. Solutions: #4: The order of (2,3) is lcm(3, 5) = 15. #14: lcm(3, 4) = 12. #18: No. When written as a direct product of prime-power cyclic groups the two representations are not identical. #20: Yes – both are isomorphic to Z2 ! Z3 ! Z4 ! Z5 ! Z9 . Page 174–176: 1, 5, 8, 11, 13, 14, 15, 29, 30, 47, 50, 55. Solutions: #8: Not a ring (this set consists of just the positive integers and it is not a group under addition.) #14: Just 1 and –1. #30: A unit in a ring R with identity is an element x for which there is a multiplicative inverse y; i.e., there exists a y such that xy = yx = 1. #50: Just show it is closed under subtraction and multiplication. Page 182–184: 1, 2, 5, 9, 11, 14, 15, 16, 29. Solutions: #2: x = 3. !1 2$ !'2 '2$ !0 0$ #14: # & # & = # & "2 4% " 1 1 % "0 0% #16: n must be the smallest such positive integer. Page 326–327: 1, 2, 3, 5, 7, 9, 13. Solution: #2: 27 Page 243–244: 3, 13. 1. The center of a ring R is the set Z(R) = {a : ar = ra for all r in R} . -
8 Algebra: Bracketsmep Y8 Practice Book A
8 Algebra: BracketsMEP Y8 Practice Book A 8.1 Expansion of Single Brackets In this section we consider how to expand (multiply out) brackets to give two or more terms, as shown below: 36318()xx+=+ First we revise negative numbers and order of operations. Example 1 Evaluate: (a) −+610 (b) −+−74() (c) ()−65×−() (d) 647×−() (e) 48()+ 3 (f) 68()− 15 ()−23−−() (g) 35−−() (h) −1 Solution (a) −+6104 = (b) −+−7474()=− − =−11 (c) ()−6530×−()= (d) 6476×−()=×−() 3 =−18 (e) 48()+ 3=× 4 11 = 44 (f) 68()− 15=×− 6() 7 =−42 127 8.1 MEP Y8 Practice Book A (g) 3535−−()=+ = 8 ()−23−−()()−23+ (h) = −1 −1 1 = −1 =−1 When a bracket is expanded, every term inside the bracket must be multiplied by the number outside the bracket. Remember to think about whether each number is positive or negative! Example 2 Expand 36()x+ using a table. Solution × x 6 3 3x 18 From the table, 36318()xx+=+ Example 3 Expand 47()x−. Solution 47()x−=×−×447x Remember that every term inside the bracket must be multiplied by =−428x the number outside the bracket. Example 4 Expand xx()8−. 128 MEP Y8 Practice Book A Solution xx()8−=×−×xxx8 =−8xx2 Example 5 Expand ()−34()− 2x. Solution ()−34()− 2x=−()34×−−() 32×x =−12 −() − 6 x =−12 + 6 x Exercises 1. Calculate: (a) −+617 (b) 614− (c) −−65 (d) 69−−() (e) −−−11() 4 (f) ()−64×−() (g) 87×−() (h) 88÷−() 4 (i) 68()− 10 (j) 53()− 10 (k) 711()− 4 (l) ()− 4617()− 2. Copy and complete the following tables, and write down each of the expansions: (a) (b) ×x 2 ×x – 7 4 5 42()x+= 57()x−= (c) (d) × x 3 ×2x 5 4 5 43()x+= 52()x+ 5= 129 8.1 MEP Y8 Practice Book A 3.