Hopf Algebras, Quantum Groups and Yang-Baxter Equations
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Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler B.S., Massachusetts Institute of Technology SUBMITTED in PA
J- Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler B.S., Massachusetts Institute of Technology (1963) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY August, 1965 Signature of Author . .-. .. Department of Mathematics, August 31, 1965 Certified by ....-.. Thesis Supervisqr Accepted by .......... 0................................... Chairman, Departmental Committee on Graduate Students v/I 2. Cocommutative Hopf Algebras 19 with Antipode by Moss Eisenberg Sweedler Submitted to the Department of Mathematics on August 31, 1965, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Abstract In the first chapter the preliminaries of the theory of Hopf algebras are presented. The notion and properties of the antipode are developed. An important filtration is induced in the Hopf algebra by its dual when the Hopf alge- bra is split. It is shown conilpotence and an algebraically closed field insure a Hopf algebra is split. The monoid of grouplike elements is studied. In the second chapter conditions for an algebra A -- which is a comodule for a Hopf algebra H --to be of the form A 'E B ® H (linear isomorphism) are given. The dual situation is studied. The graded Hopf algebra associated with a split Hopf algebra decomposes in the above manner. Chapter III contains the cohomology theory of a commutative algebra which is a module for a cocommutative Hopf algebra. There is extension theory and specialization to the situation the Hopf algebra is a group algebra. Chapter IV is dual to chapter III. Chapter V is devoted to coconnected cocommutative Hopf algebras, mostly in characteristic p > 0 . -
Integrable Systems and Quantum Deformations
Integrable Systems and Quantum Deformations Peter Koroteev University of Minnesota In collaboration with N. Beisert, F. Spill, A. Rej 0802.0777, work in progress MG12, July 13th 2009 Outline Introduction. Integrability and Symmetry What is Integrability Coordinate Bethe Ansatz AdS/CFT Correspondence Quantum Deformation of 1D Hubbard Model 2 Uq(su(2j2) n R ) algebra Hubbard-like Models Universal R-matrix Yangian Universal R-matrix I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I Integrable models: not necessrily easy/complicated Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable Integrable models can be completely solved Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Why Integrability? I Easy models: Classical Mechanics (ossilator, free point particle) I \Complicated", Chaotic models. Almost untractable I Integrable models: not necessrily easy/complicated Integrable models can be completely solved I Finite dimensional: mechanical with n d.o.f.s { exist n-1 local integrals in involution I Field Theory: Infinite dimensional symmetry, S{matrix satisfies integrability constraints. We are -
Forms of Hopf Algebras and Galois Theory 3
FORMS OF HOPF ALGEBRAS AND GALOIS THEORY BODO PAREIGIS The theory of Hopf algebras is closely connected with various ap- plications, in particular to algebraic and formal groups. Although the rst o ccurence of Hopf algebras was in algebraic top ology, they are now found in areas as remote as combinatorics and analysis. Their struc- ture has b een studied in great detail and many of their prop erties are well understo o d. We are interested in a systematic treatmentofHopf algebras with the techniques of forms and descent. The rst three paragraphs of this pap er givea survey of the present state of the theory of forms of Hopf algebras and of Hopf Galois theory esp ecially for separable extensions. It includes many illustrating exam- ples some of which cannot b e found in detail in the literature. The last two paragraphs are devoted to some new or partial results on the same eld. There we formulate some of the op en questions whichshould be interesting ob jects for further study.We assume throughout most of the pap er that k is a base eld and do not touch up on the recent b eautiful results of Hopf Galois theory for rings of integers in algebraic numb er elds as develop ed in [C1]. 1. Hopf algebra forms As a rst example of the o ccurence of a Hopf algebra let us consider the units functor. In the sequel let k be a commutative, asso ciativering with unit. Later on it will b e a eld, in particular the eld of rationals or reals. -
Hopf Rings in Algebraic Topology
HOPF RINGS IN ALGEBRAIC TOPOLOGY W. STEPHEN WILSON Abstract. These are colloquium style lecture notes about Hopf rings in al- gebraic topology. They were designed for use by non-topologists and graduate students but have been found helpful for those who want to start learning about Hopf rings. They are not “up to date,” nor are then intended to be, but instead they are intended to be introductory in nature. Although these are “old” notes, Hopf rings are thriving and these notes give a relatively painless introduction which should prepare the reader to approach the current litera- ture. This is a brief survey about Hopf rings: what they are, how they arise, examples, and how to compute them. There are very few proofs. The bulk of the technical details can be found in either [RW77] or [Wil82], but a “soft” introduction to the material is difficult to find. Historically, Hopf algebras go back to the early days of our subject matter, homotopy theory and algebraic topology. They arise naturally from the homology of spaces with multiplications on them, i.e. H-spaces, or “Hopf” spaces. In our language, this homology is a group object in the category of coalgebras. Hopf algebras have become objects of study in their own right, e.g. [MM65] and [Swe69]. They were also able to give great insight into complicated structures such as with Milnor’s work on the Steenrod algebra [Mil58]. However, when spaces have more structure than just a multiplication, their homology produces even richer algebraic stuctures. In particular, with the development of generalized cohomology theories, we have seen that the spaces which classify them have a structure mimicking that of a graded ring. -
Simultaneous Diagonalization and SVD of Commuting Matrices
Simultaneous Diagonalization and SVD of Commuting Matrices Ronald P. Nordgren1 Brown School of Engineering, Rice University Abstract. We present a matrix version of a known method of constructing common eigenvectors of two diagonalizable commuting matrices, thus enabling their simultane- ous diagonalization. The matrices may have simple eigenvalues of multiplicity greater than one. The singular value decomposition (SVD) of a class of commuting matrices also is treated. The effect of row/column permutation is examined. Examples are given. 1 Introduction It is well known that if two diagonalizable matrices have the same eigenvectors, then they commute. The converse also is true and a construction for the common eigen- vectors (enabling simultaneous diagonalization) is known. If one of the matrices has distinct eigenvalues (multiplicity one), it is easy to show that its eigenvectors pertain to both commuting matrices. The case of matrices with simple eigenvalues of mul- tiplicity greater than one requires a more complicated construction of their common eigenvectors. Here, we present a matrix version of a construction procedure given by Horn and Johnson [1, Theorem 1.3.12] and in a video by Sadun [4]. The eigenvector construction procedure also is applied to the singular value decomposition of a class of commuting matrices that includes the case where at least one of the matrices is real and symmetric. In addition, we consider row/column permutation of the commuting matrices. Three examples illustrate the eigenvector construction procedure. 2 Eigenvector Construction Let A and B be diagonalizable square matrices that commute, i.e. AB = BA (1) and their Jordan canonical forms read −1 −1 A = SADASA , B = SBDB SB . -
Lie Algebras and Representation Theory Andreasˇcap
Lie Algebras and Representation Theory Fall Term 2016/17 Andreas Capˇ Institut fur¨ Mathematik, Universitat¨ Wien, Nordbergstr. 15, 1090 Wien E-mail address: [email protected] Contents Preface v Chapter 1. Background 1 Group actions and group representations 1 Passing to the Lie algebra 5 A primer on the Lie group { Lie algebra correspondence 8 Chapter 2. General theory of Lie algebras 13 Basic classes of Lie algebras 13 Representations and the Killing Form 21 Some basic results on semisimple Lie algebras 29 Chapter 3. Structure theory of complex semisimple Lie algebras 35 Cartan subalgebras 35 The root system of a complex semisimple Lie algebra 40 The classification of root systems and complex simple Lie algebras 54 Chapter 4. Representation theory of complex semisimple Lie algebras 59 The theorem of the highest weight 59 Some multilinear algebra 63 Existence of irreducible representations 67 The universal enveloping algebra and Verma modules 72 Chapter 5. Tools for dealing with finite dimensional representations 79 Decomposing representations 79 Formulae for multiplicities, characters, and dimensions 83 Young symmetrizers and Weyl's construction 88 Bibliography 93 Index 95 iii Preface The aim of this course is to develop the basic general theory of Lie algebras to give a first insight into the basics of the structure theory and representation theory of semisimple Lie algebras. A problem one meets right in the beginning of such a course is to motivate the notion of a Lie algebra and to indicate the importance of representation theory. The simplest possible approach would be to require that students have the necessary background from differential geometry, present the correspondence between Lie groups and Lie algebras, and then move to the study of Lie algebras, which are easier to understand than the Lie groups themselves. -
The Algebra Generated by Three Commuting Matrices
THE ALGEBRA GENERATED BY THREE COMMUTING MATRICES B.A. SETHURAMAN Abstract. We present a survey of an open problem concerning the dimension of the algebra generated by three commuting matrices. This article concerns a problem in algebra that is completely elementary to state, yet, has proven tantalizingly difficult and is as yet unsolved. Consider C[A; B; C] , the C-subalgebra of the n × n matrices Mn(C) generated by three commuting matrices A, B, and C. Thus, C[A; B; C] consists of all C- linear combinations of \monomials" AiBjCk, where i, j, and k range from 0 to infinity. Note that C[A; B; C] and Mn(C) are naturally vector-spaces over C; moreover, C[A; B; C] is a subspace of Mn(C). The problem, quite simply, is this: Is the dimension of C[A; B; C] as a C vector space bounded above by n? 2 Note that the dimension of C[A; B; C] is at most n , because the dimen- 2 sion of Mn(C) is n . Asking for the dimension of C[A; B; C] to be bounded above by n when A, B, and C commute is to put considerable restrictions on C[A; B; C]: this is to require that C[A; B; C] occupy only a small portion of the ambient Mn(C) space in which it sits. Actually, the dimension of C[A; B; C] is already bounded above by some- thing slightly smaller than n2, thanks to a classical theorem of Schur ([16]), who showed that the maximum possible dimension of a commutative C- 2 subalgebra of Mn(C) is 1 + bn =4c. -
Arxiv:1705.10957V2 [Math.AC] 14 Dec 2017 Ideal Called Is and Commutator X Y Vrafield a Over Let Matrix Eae Leri Es Is E Sdfierlvn Notions
NEARLY COMMUTING MATRICES ZHIBEK KADYRSIZOVA Abstract. We prove that the algebraic set of pairs of matrices with a diag- onal commutator over a field of positive prime characteristic, its irreducible components, and their intersection are F -pure when the size of matrices is equal to 3. Furthermore, we show that this algebraic set is reduced and the intersection of its irreducible components is irreducible in any characteristic for pairs of matrices of any size. In addition, we discuss various conjectures on the singularities of these algebraic sets and the system of parameters on the corresponding coordinate rings. Keywords: Frobenius, singularities, F -purity, commuting matrices 1. Introduction and preliminaries In this paper we study algebraic sets of pairs of matrices such that their commutator is either nonzero diagonal or zero. We also consider some other related algebraic sets. First let us define relevant notions. Let X =(x ) and Y =(y ) be n×n matrices of indeterminates arXiv:1705.10957v2 [math.AC] 14 Dec 2017 ij 1≤i,j≤n ij 1≤i,j≤n over a field K. Let R = K[X,Y ] be the polynomial ring in {xij,yij}1≤i,j≤n and let I denote the ideal generated by the off-diagonal entries of the commutator matrix XY − YX and J denote the ideal generated by the entries of XY − YX. The ideal I defines the algebraic set of pairs of matrices with a diagonal commutator and is called the algebraic set of nearly commuting matrices. The ideal J defines the algebraic set of pairs of commuting matrices. -
Structure Theory of Finite Conformal Algebras Alessandro D'andrea JUN
Structure theory of finite conformal algebras by Alessandro D'Andrea Laurea in Matematica, Universith degli Studi di Pisa (1994) Diploma, Scuola Normale Superiore (1994) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1998 @Alessandro D'Andrea, 1998. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. A uthor .. ................................ Department of Mathematics May 6, 1998 Certified by ............................ Victor G. Kac Professor of Mathematics rc7rc~ ~ Thesis Supervisor Accepted by. Richard B. Melrose ,V,ASSACHUSETT S: i i. Chairman, Department Committee OF TECHNOLCaY JUN 0198 LIBRARIES Structure theory of finite conformal algebras by Alessandro D'Andrea Submitted to the Department ,of Mathematics on May 6, 1998, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis I gave a classification of simple and semi-simple conformal algebras of finite rank, and studied their representation theory, trying to prove or disprove the analogue of the classical Lie algebra representation theory results. I re-expressed the operator product expansion (OPE) of two formal distributions by means of a generating series which I call "A-bracket" and studied the properties of the resulting algebraic structure. The above classification describes finite systems of pairwise local fields closed under the OPE. Thesis Supervisor: Victor G. Kac Title: Professor of Mathematics Acknowledgments The few people I would like to thank are those who delayed my thesis the most. -
Algebra & Number Theory
Algebra & Number Theory Volume 7 2013 No. 7 Hopf monoids from class functions on unitriangular matrices Marcelo Aguiar, Nantel Bergeron and Nathaniel Thiem msp ALGEBRA AND NUMBER THEORY 7:7 (2013) msp dx.doi.org/10.2140/ant.2013.7.1743 Hopf monoids from class functions on unitriangular matrices Marcelo Aguiar, Nantel Bergeron and Nathaniel Thiem We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal’s category of species. Such structure is carried by the collection of class function spaces on those groups and also by the collection of superclass function spaces in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices. Introduction A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. Combinatorial structures that compose and decompose give rise to Hopf monoids. These objects are the subject of[Aguiar and Mahajan 2010, Part II]. The few basic notions and examples needed for our purposes are reviewed in Section 1, including the Hopf monoids of linear orders, set partitions, and simple graphs and the Hadamard product of Hopf monoids. -
Lecture 2: Spectral Theorems
Lecture 2: Spectral Theorems This lecture introduces normal matrices. The spectral theorem will inform us that normal matrices are exactly the unitarily diagonalizable matrices. As a consequence, we will deduce the classical spectral theorem for Hermitian matrices. The case of commuting families of matrices will also be studied. All of this corresponds to section 2.5 of the textbook. 1 Normal matrices Definition 1. A matrix A 2 Mn is called a normal matrix if AA∗ = A∗A: Observation: The set of normal matrices includes all the Hermitian matrices (A∗ = A), the skew-Hermitian matrices (A∗ = −A), and the unitary matrices (AA∗ = A∗A = I). It also " # " # 1 −1 1 1 contains other matrices, e.g. , but not all matrices, e.g. 1 1 0 1 Here is an alternate characterization of normal matrices. Theorem 2. A matrix A 2 Mn is normal iff ∗ n kAxk2 = kA xk2 for all x 2 C : n Proof. If A is normal, then for any x 2 C , 2 ∗ ∗ ∗ ∗ ∗ 2 kAxk2 = hAx; Axi = hx; A Axi = hx; AA xi = hA x; A xi = kA xk2: ∗ n n Conversely, suppose that kAxk = kA xk for all x 2 C . For any x; y 2 C and for λ 2 C with jλj = 1 chosen so that <(λhx; (A∗A − AA∗)yi) = jhx; (A∗A − AA∗)yij, we expand both sides of 2 ∗ 2 kA(λx + y)k2 = kA (λx + y)k2 to obtain 2 2 ∗ 2 ∗ 2 ∗ ∗ kAxk2 + kAyk2 + 2<(λhAx; Ayi) = kA xk2 + kA yk2 + 2<(λhA x; A yi): 2 ∗ 2 2 ∗ 2 Using the facts that kAxk2 = kA xk2 and kAyk2 = kA yk2, we derive 0 = <(λhAx; Ayi − λhA∗x; A∗yi) = <(λhx; A∗Ayi − λhx; AA∗yi) = <(λhx; (A∗A − AA∗)yi) = jhx; (A∗A − AA∗)yij: n ∗ ∗ n Since this is true for any x 2 C , we deduce (A A − AA )y = 0, which holds for any y 2 C , meaning that A∗A − AA∗ = 0, as desired. -
A Gentle Introduction to a Beautiful Theorem of Molien
A Gentle Introduction to a Beautiful Theorem of Molien Holger Schellwat [email protected], Orebro¨ universitet, Sweden Universidade Eduardo Mondlane, Mo¸cambique 12 January, 2017 Abstract The purpose of this note is to give an accessible proof of Moliens Theorem in Invariant Theory, in the language of today’s Linear Algebra and Group Theory, in order to prevent this beautiful theorem from being forgotten. Contents 1 Preliminaries 3 2 The Magic Square 6 3 Averaging over the Group 9 4 Eigenvectors and eigenvalues 11 5 Moliens Theorem 13 6 Symbol table 17 7 Lost and found 17 References 17 arXiv:1701.04692v1 [math.GM] 16 Jan 2017 Index 18 1 Introduction We present some memories of a visit to the ring zoo in 2004. This time we met an animal looking like a unicorn, known by the name of invariant theory. It is rare, old, and very beautiful. The purpose of this note is to give an almost self contained introduction to and clarify the proof of the amazing theorem of Molien, as presented in [Slo77]. An introduction into this area, and much more, is contained in [Stu93]. There are many very short proofs of this theorem, for instance in [Sta79], [Hu90], and [Tam91]. Informally, Moliens Theorem is a power series generating function formula for counting the dimensions of subrings of homogeneous polynomials of certain degree which are invariant under the action of a finite group acting on the variables. As an apetizer, we display this stunning formula: 1 1 ΦG(λ) := |G| det(id − λTg) g∈G X We can immediately see elements of linear algebra, representation theory, and enumerative combinatorics in it, all linked together.