DOCSLIB.ORG

An Introduction to Quantum Symmetries Réamonn´O Buachalla

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Quantum Symmetries Réamonn´O Buachalla An Introduction to Quantum Symmetries Lectures by: R´eamonn O´ Buachalla (Notes by: Fatemeh Khosravi) Noncommutative Geometry the Next Generation (19th September-14th October 2016 ) 10:00 - 10:45 Institute of Mathematics Polish Academy of Science(IMPAN) Warsaw 1 Contents 1 From Lie Algebras to Hopf Algebras 3 1.1 Lie Algebras . 3 1.2 Quick Summary of Lie Groups . 4 1.3 Universal Enveloping Algebras . 5 1.4 Coalgebras and Bialgebras . 7 1.5 Universal Enveloping Algebras as Bialgebras . 8 1.6 Hopf Algebras . 9 1.7 Sweedler Notation . 10 1.8 Properties of the Antipode . 11 2 q-Deforming the Hopf Algebra U(sl2) 12 2.1 The Hopf Algebra Uq(sl2)........................... 13 2.2 The Classical (q = 1)-Limit of Uq(sl2) .................... 14 2.3 Representation Theory . 15 2.4 q-Integers . 16 2.5 The Representations T!;l ............................ 16 2.6 The Generic Case . 17 2.7 The Root of Unity Case . 17 3 Algebraic Groups and Commutative Hopf Algebras 18 3.1 Algebraic Sets and Radical Ideals . 18 3.2 Morphisms . 19 3.3 Algebraic Groups . 20 3.4 Hopf Algebras . 21 3.5 The Hopf Algebra Oq(SLn).......................... 21 4 Finite Duals and the Peter{Weyl Theorem for Oq(SL2) 22 4.1 The Hopf Dual of a Hopf Algebra . 22 4.2 Quantum Coordinate Functions . 24 4.3 A q-Deformed Peter{Weyl Theorem . 25 4.4 Dual Pairings of Hopf Algebras . 25 4.5 Comodules and Dual Pairings . 26 5 Cosemisimplicity and Compact Quantum Groups 27 5.1 Cosemisimple Coalgebras . 27 2 5.2 Compact Quantum Group Algebras . 30 5.3 Compact Quantum Groups . 31 5.3.1 Compact Groups . 31 5.3.2 C∗-algebras and the Gelfand{Naimark Theorem . 31 5.3.3 Compact Quantum Semigroups . 32 5.3.4 Compact Quantum Groups . 33 5.4 The Koornwinder-Dijkhuizen Correspondence . 34 5.4.1 The Haar State . 34 6 Principal Comodule Algebras 35 6.1 Hopf{Galois Extensions . 35 6.2 Quantum Homogeneous Spaces . 36 6.3 The Podle´sSphere . 36 6.4 Faithfull Flatness . 38 6.4.1 General Definition . 38 6.4.2 Quantum Homogeneous Spaces and Takeuchi's Equivalence . 38 6.5 Strong Connections and Principal Comodule Algebras . 39 6.5.1 First-Order Differential Calculi . 40 6.6 Universal Quantum Principal Bundle . 40 6.6.1 Strong Connections and Quantum Principal Comodule Algebras . 41 6.6.2 Connections for Associated Bundles . 41 6.7 The Podel´sSphere . 43 7 Differential Calculi, Complex Structures, and K¨ahler{Dirac Operators 43 7.1 Classifying first-order differential calculus on a Quantum Homogeneous Space ...................................... 43 7.2 The Podle´sSphere . 44 7.3 Complexes and Double Complexes . 45 7.4 Differential Calculi . 45 7.5 Complex Structures . 45 7.6 Hermitian Surfaces and K¨ahler{Dirac Operators . 46 7.6.1 Hermitian Surfaces . 46 7.7 A Spectral Triple for the Podle´sSphere . 47 8 Appendix 47 8.1 Two Different Definitions of Compact Quantum Qroup . 47 3 1 From Lie Algebras to Hopf Algebras 1.1 Lie Algebras We begin with some basic conventions: Unless stated otherwise, all algebras discussed here will be assumed to be unital, and all algebras, and anti-algebras, maps will be assumed to be unital. Moreover, all vector spaces V will be assumed to be over C. Definition 1.1. A Lie algebra is a vector space g together with a bilinear map [·; ·]: g × g ! g, called the Lie bracket, satisfying the following properties: 1. [x; y] = −[y; z], for all x; y 2 g, 2. (Jacobi Identity)[x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0, for all x; y; z 2 g. A Lie subalgebra h of a Lie algebra g is a subspace of g which is closed with respect to the Lie bracket. Example 1.2. The zero vector space, together with the zero bilinear map is a Lie algebra. We call it the 0 Lie algebra. Definition 1.3. A Lie algebra g is said to be abelian or commutative if [x; y] = 0; for all x; y 2 g: Given a vector space V, let gl(V ) denote the Lie algebra enveloped by the associative algebra of all linear endomorphisms of V . A representation of a Lie algebra g on V is a Lie algebra homomorphism π : g ! gl(V ): Example 1.4. Show that the matrices g = Mn(C), together with the usual commuta- tion bracket, form a Lie algebra. Definition 1.5. A derivation on an algebra A is a linear map D : A ! A such that D(ab) = D(a)b + aD(b) a; b 2 A Exercise 1.6. For an algebra A consider the subspace of gl(A) consisting of all deriva- tions on A. Show that this is a Lie subalgebra of gl(A). Exercise 1.7. More generally, show that any associative algebra A can be given the structure of a Lie algebra by defining a Lie bracket [a; b] := ab − ba; a; b 2 A: We denote this Lie algebra by L(A). 4 It is easy to see that for any associative algebra A, its Lie algebra L(A) is abelian if and only if A is abelian as an associative algebra. Now we introduce a Lie algebra which its bracket does not arise from an algebra product. 3 Example 1.8. Assume R as a vector space over R. Then the vector product (x; y) 7! 3 x ^ y defines the structure of a Lie algebra on R . Explicitly, if x = (x1; x2; x3) and (y1; y2; y3), then x ^ y = (x2y3 − x3y2; x3y1 − x1y3; x1y2 − x2y1) Exercise 1.9. The Lie algebra sl2 is the vector space spanned by the three elements E; H, and F together with the Lie bracket defined by [E; F ] = H; [H; E] = 2E; [H; F ] = −2F: Show that this is a Lie bracket, and show that an injection of Lie algebras is give by sl2 ,! M3(C); where 0 1 0 0 1 0 E 7! ;F 7! ;H 7! : 0 0 1 0 0 −1 Definition 1.10. A homomorphism between two Lie algebras is a linear map that is compatible with the respective Lie brackets: ρ : g ! g0; ρ([x; y]) = [ρ(x); ρ(y)]; for all elements x; y 2 g 1.2 Quick Summary of Lie Groups Definition 1.11. A Lie group is a group that is also a manifold G such that G × G ! GG ! G −1 (g1; g2) 7! g1g2 g 7! g are smooth maps. Note. For a Lie group G, there exists an action G × C1(G) ! C1(G) such that (g ◦ F )(g0) = F (g−1g0) for all g; g0 2 G and F 2 C1(G). n 1 n 1 n Definition 1.12. A vector field on R is a derivation D : C (R ) ! C (R ). In general a vector field on a manifold M is a derivation D : C1(M) ! C1(M). Exercise 1.13. Let Der(M) denote the set of all derivations D : C1(M) ! C1(M), then there exists a Lie bracket on Der(M). [X; Y ] = X ◦ Y − Y ◦ X 5 Definition 1.14. A left-invariant vector field on a Lie group is a derivation which is a left G-module map. Fact. For a Lie group G, the Lie algebra (Der(G); [·; ·]) := Lie(G) is finite dimensional. We call it the Lie algebra of G. A simple consequence of Ado's theorem (that every finite dimensional real Lie algebra can be embedded in gl(V ), for some V ) is the following. Theorem 1.15 Every finite dimensional real Lie algebra is a Lie algebra of a Lie group. Example 1.16. sl2 is the Lie algebra of SL(2). 1.3 Universal Enveloping Algebras In this subsection we introduce tensor products and the universal enveloping algebra of a Lie algebra. These objects are defined using universal properties, which if you are comfortable enough with category theory, you might want to look up the general definition of. Lemma 1.17 Given two vector spaces U and V there exists a unique vector space U ⊗V and a bilinear map ' : U × V ! U ⊗ V such that, for any bilinear map f : U × V ! W , for some vector space W , there exists a unique linear map fe : U ⊗ V ! W such that the following diagram commutes i U × V / U ⊗ V fe f * W: Exercise 1.18. Prove the lemma by concluding uniqueness directly and by concretely constructing U ⊗ V . Hint: Take the vector space with basis given by the elements of U × V and find a suitable quotient. Definition 1.19. The tensor algebra of a vector space V is the vector space 1 M T (V ) := V ⊗k; k=0 0 where we use the convention V = C. If A and B are two algebras a multiplication can be defined on A ⊗ B by (a ⊗ b):(a0 ⊗ b0) := aa0 ⊗ bb0; a; a0 2 A; b; b0 2 B: 6 Endowed with this multiplication we call A ⊗ B the algebra tensor product of A and B. We now come to the universal enveloping algebra of a Lie algebra, also defined using a universal property Lemma 1.20 For any Lie algebra g there exists a unique pair (U(g); i) (up to isomor- phism) where U(g) is a unital associative algebra and i : g !L(U(g)) is a Lie algebra ho- momorphism such that for any associative algebra A and Lie algebra homomorphism f : g !L(A), there exists a unique homomorphism of associative algebras fe : U(g) ! A such that the following diagram of linear maps is commutative i g / U(g) fe f ) A: Proof.
Load more

Recommended publications
  • NOTES on MODULES and ALGEBRAS 1. Some Remarks About Categories Throughout These Notes, We Will Be Using Some of the Basic Termin
    NOTES ON MODULES AND ALGEBRAS WILLIAM SCHMITT 1. Some Remarks about Categories Throughout these notes, we will be using some of the basic terminol- ogy and notation from category theory. Roughly speaking, a category C consists of some class of mathematical structures, called the objects of C, together with the appropriate type of mappings between objects, called the morphisms, or arrows, of C. Here are a few examples: Category Objects Morphisms Set sets functions Grp groups group homomorphisms Ab abelian groups group homomorphisms ModR left R-modules (where R-linear maps R is some fixed ring) For objects A and B belonging to a category C we denote by C(A, B) the set of all morphisms from A to B in C and, for any f ∈ C(A, B), the objects A and B are called, respectively, the domain and codomain of f. For example, if A and B are abelian groups, and we denote by U(A) and U(B) the underlying sets of A and B, then Ab(A, B) is the set of all group homomorphisms having domain A and codomain B, while Set(U(A), U(B)) is the set of all functions from A to B, where the group structure is ignored. Whenever it is understood that A and B are objects in some category C then by a morphism from A to B we shall always mean a morphism from A to B in the category C. For example if A and B are rings, a morphism f : A → B is a ring homomorphism, not just a homomorphism of underlying additive abelian groups or multiplicative monoids, or a function between underlying sets.
    [Show full text]
  • Topics in Module Theory
    Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order.
    [Show full text]
  • Embeddings of Integral Quadratic Forms Rick Miranda Colorado State
    Embeddings of Integral Quadratic Forms Rick Miranda Colorado State University David R. Morrison University of California, Santa Barbara Copyright c 2009, Rick Miranda and David R. Morrison Preface The authors ran a seminar on Integral Quadratic Forms at the Institute for Advanced Study in the Spring of 1982, and worked on a book-length manuscript reporting on the topic throughout the 1980’s and early 1990’s. Some new results which are proved in the manuscript were announced in two brief papers in the Proceedings of the Japan Academy of Sciences in 1985 and 1986. We are making this preliminary version of the manuscript available at this time in the hope that it will be useful. Still to do before the manuscript is in final form: final editing of some portions, completion of the bibliography, and the addition of a chapter on the application to K3 surfaces. Rick Miranda David R. Morrison Fort Collins and Santa Barbara November, 2009 iii Contents Preface iii Chapter I. Quadratic Forms and Orthogonal Groups 1 1. Symmetric Bilinear Forms 1 2. Quadratic Forms 2 3. Quadratic Modules 4 4. Torsion Forms over Integral Domains 7 5. Orthogonality and Splitting 9 6. Homomorphisms 11 7. Examples 13 8. Change of Rings 22 9. Isometries 25 10. The Spinor Norm 29 11. Sign Structures and Orientations 31 Chapter II. Quadratic Forms over Integral Domains 35 1. Torsion Modules over a Principal Ideal Domain 35 2. The Functors ρk 37 3. The Discriminant of a Torsion Bilinear Form 40 4. The Discriminant of a Torsion Quadratic Form 45 5.
    [Show full text]
  • Banach Spaces of Distributions Having Two Module Structures
    JOURNAL OF FUNCTIONAL ANALYSIS 51, 174-212 (1983) Banach Spaces of Distributions Having Two Module Structures W. BRAUN Institut fiir angewandte Mathematik, Universitiir Heidelberg, Im Neuheimer Feld 294, D-69 Heidelberg, Germany AND HANS G. FEICHTINGER Institut fir Mathematik, Universitiit Wien, Strudlhofgasse 4, A-1090 Vienna, Austria Communicated by Paul Malliavin Received November 1982 After a discussion of a space of test functions and the correspondingspace of distributions, a family of Banach spaces @,I[ [Is) in standardsituation is described. These are spaces of distributions having a pointwise module structure and also a module structure with respect to convolution. The main results concern relations between the different spaces associated to B established by means of well-known methods from the theory of Banach modules, among them B, and g, the closure of the test functions in B and the weak relative completion of B, respectively. The latter is shown to be always a dual Banach space. The main diagram, given in Theorem 4.7, gives full information concerning inclusions between these spaces, showing also a complete symmetry. A great number of corresponding formulas is established. How they can be applied is indicated by selected examples, in particular by certain Segal algebras and the A,-algebras of Herz. Various further applications are to be given elsewhere. 0. 1~~RoDucT10N While proving results for LP(G), 1 < p < co for a locally compact group G, one sometimes does not need much more information about these spaces than the fact that L’(G) is an essential module over Co(G) with respect to pointwise multiplication and an essential L’(G)-module for convolution.
    [Show full text]
  • The Geometry of Lubin-Tate Spaces (Weinstein) Lecture 1: Formal Groups and Formal Modules
    The Geometry of Lubin-Tate spaces (Weinstein) Lecture 1: Formal groups and formal modules References. Milne's notes on class field theory are an invaluable resource for the subject. The original paper of Lubin and Tate, Formal Complex Multiplication in Local Fields, is concise and beautifully written. For an overview of Dieudonn´etheory, please see Katz' paper Crystalline Cohomology, Dieudonn´eModules, and Jacobi Sums. We also recommend the notes of Brinon and Conrad on p-adic Hodge theory1. Motivation: the local Kronecker-Weber theorem. Already by 1930 a great deal was known about class field theory. By work of Kronecker, Weber, Hilbert, Takagi, Artin, Hasse, and others, one could classify the abelian extensions of a local or global field F , in terms of data which are intrinsic to F . In the case of global fields, abelian extensions correspond to subgroups of ray class groups. In the case of local fields, which is the setting for most of these lectures, abelian extensions correspond to subgroups of F ×. In modern language there exists a local reciprocity map (or Artin map) × ab recF : F ! Gal(F =F ) which is continuous, has dense image, and has kernel equal to the connected component of the identity in F × (the kernel being trivial if F is nonarchimedean). Nonetheless the situation with local class field theory as of 1930 was not completely satisfying. One reason was that the construction of recF was global in nature: it required embedding F into a global field and appealing to the existence of a global Artin map. Another problem was that even though the abelian extensions of F are classified in a simple way, there wasn't any explicit construction of those extensions, save for the unramified ones.
    [Show full text]
  • Symmetric Invariant Bilinear Forms on Modular Vertex Algebras
    Symmetric Invariant Bilinear Forms on Modular Vertex Algebras Haisheng Li Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA, and School of Mathematical Sciences, Xiamen University, Xiamen 361005, China E-mail address: [email protected] Qiang Mu School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China E-mail address: [email protected] September 27, 2018 Abstract In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic p). We first introduce a bialgebra H and we then introduce a notion of H-module vertex algebra and a notion of (V, H)-module for an H-module vertex algebra V . Then we give a modular version of Frenkel-Huang-Lepowsky’s theory and study invariant bilinear forms on an H-module vertex algebra. As the main results, we obtain an explicit description of the space of invariant bilinear forms on a general H-module vertex algebra, and we apply our results to affine vertex algebras and Virasoro vertex algebras. 1 Introduction arXiv:1711.00986v1 [math.QA] 3 Nov 2017 In the theory of vertex operator algebras in characteristic zero, an important notion is that of con- tragredient dual (module), which was due to Frenkel, Huang, and Lepowsky (see [FHL]). Closely related to contragredient dual is the notion of invariant bilinear form on a vertex operator algebra and it was proved therein that every invariant bilinear form on a vertex operator algebra is au- tomatically symmetric. Contragredient dual and symmetric invariant bilinear forms have played important roles in various studies. Note that as part of its structure, a vertex operator algebra V is a natural module for the Virasoro algebra.
    [Show full text]
  • Module Theory: an Approach to Linear Algebra
    MODULE THEORY An approach to linear algebra Electronic Edition T. S. Blyth Professor Emeritus, University of St Andrews 0 0 0 ? ? ? ? ? ? ? ? ? y y y 0 A B B B=(A B) 0 −−−−−−−! \? −−−−−−−−!?−−−−−−−! ?\ −−−−−! ? ? ? ? ? ? y y y 0 A M M=A 0 −−−−−−−−!?−−−−−−−−−! ? −−−−−−−−! ? −−−−−−−! ? ? ? ? ? ? y y y 0 A=(A B) M=B M=(A + B) 0 ? ? ? −−−−−! ?\ −−−−−! ? −−−−−! ? −−−−−! ? ? ? y y y 0 0 0 PREFACE to the Second O.U.P. Edition 1990 Many branches of algebra are linked by the theory of modules. Since the notion of a module is obtained essentially by a modest generalisation of that of a vector space, it is not surprising that it plays an important role in the theory of linear algebra. Modules are also of great importance in the higher reaches of group theory and ring theory, and are fundamental to the study of advanced topics such as homological algebra, category theory, and algebraic topology. The aim of this text is to develop the basic properties of modules and to show their importance, mainly in the theory of linear algebra. The first eleven sections can easily be used as a self-contained course for first year honours students. Here we cover all the basic material on modules and vector spaces required for embarkation on advanced courses. Concerning the prerequisite algebraic background for this, we mention that any standard course on groups, rings, and fields will suffice. Although we have kept the discussion as self-contained as pos- sible, there are places where references to standard results are unavoidable; readers who are unfamiliar with such results should consult a standard text on abstract alge- bra.
    [Show full text]
  • Specialization of Quadratic and Symmetric Bilinear Forms
    Manfred Knebusch Specialization of Quadratic and Symmetric Bilinear Forms Translated from German by Thomas Unger Preliminary Version (22 Nov 2009) Page: 1 job: book macro: PMONO01 date/time: 15-Dec-2009/14:12 Page: 2 job: book macro: PMONO01 date/time: 15-Dec-2009/14:12 Dedicated to the memory of my teachers Emil Artin 1898 { 1962 Hel Braun 1914 { 1986 Ernst Witt 1911 { 1991 Page: V job: book macro: PMONO01 date/time: 15-Dec-2009/14:12 Page: VI job: book macro: PMONO01 date/time: 15-Dec-2009/14:12 Preface A Mathematician Said Who Can Quote Me a Theorem that's True? For the ones that I Know Are Simply not So, When the Characteristic is Two! This pretty limerick first came to my ears in May 1998 during a talk by T.Y. Lam on field invariants from the theory of quadratic forms.1 It is { poetic exaggeration allowed { a suitable motto for this monograph. What is it about? In the beginning of the seventies I drew up a special- ization theory of quadratic and symmetric bilinear forms over fields [K4]. Let λ: K ! L [ 1 be a place. Then one can assign a form λ∗(') to a form ' over K in a meaningful way if ' has \good reduction" with respect to λ (see x1). The basic idea is to simply apply the place λ to the coefficients of ' which therefore of course have to be in the valuation ring of λ. The specialization theory of that time was satisfactory as long as the field L, and therefore also K, had characteristic 6= 2.
    [Show full text]
  • LIE ALGEBRAS: LECTURE 5 27 April 2010 1. Modules Let L Be a Lie
    LIE ALGEBRAS: LECTURE 5 27 April 2010 CRYSTAL HOYT 1. Modules Let L be a Lie algebra. A vector space V with an operation L × V ! V (x; v) 7! x:v is called an L-module if for all x; y 2 L, v; w 2 V , a; b 2 F the following conditions hold: (1) (ax + by).v = a(x.v) + b(y.v) (2) x.(av + bw) = a(x.v) + b(x.w) (3) [xy].v = x.y.v - y.x.v. Example 1.1. If φ : L ! gl(V ) is a representation of L, then V is an L- module via the action x:v = φ(x)v. Conversely, if V is an L-module then φ(x)v = x:v defines a representation. Fix a Lie algebra L, and let V and W be L-modules. An L-module homo- morphism is a linear map f : V ! W such that f(x:v) = x:f(v) for all x 2 L and v 2 V . The kernel of this map is an L-submodule of V . If f : V ! W is an L-module homomorphism and is an isomorphism of vector spaces, then f −1 : W ! V is also an L-module homomorphism. In this case, we call f an isomorphism of L-modules, and the modules V and W are called equivalent representations of L. An L-module V is called irreducible (or simple) if it has precisely two L- submodules: itself and 0. A direct sum of L-modules V1;:::;Vt is a direct sum of vector spaces V1 ⊕ · · · ⊕ Vt, with the action of L defined: x:(v1; v2; : : : ; vt) = (x:v1; x:v2; : : : ; x:vt): 1 An L-module V is called completely reducible (or semisimple) if V is a direct sum of irreducible of L-submodules.
    [Show full text]
  • Arxiv:Math/0608781V2 [Math.QA] 10 Apr 2007 Lorfrne Hri.W Ee to Refer We Fus Therein
    ON MODULE CATEGORIES OVER FINITE-DIMENSIONAL HOPF ALGEBRAS NICOLAS´ ANDRUSKIEWITSCH AND JUAN MART´IN MOMBELLI Abstract. We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to equi- variant Morita equivalence. Specifically, any indecomposable exact module categories is equivalent to the category of finite-dimensional modules over a left comodule algebra. This is an alternative approach to the results of Etingof and Ostrik. For this, we study the stabilizer introduced by Yan and Zhu and show that it coincides with the internal Hom. We also describe the correspondence of module categories between Rep H and Rep(H∗). Contents Introduction 1 1. Module categories 2 2. Yan-Zhu Stabilizers 14 3. Applications of the Yan-Zhu stabilizers to module categories 27 References 33 Introduction The notion of fusion category is a far-reaching generalization of the notion of finite group. It has been studied in connection with different problems in conformal field the- arXiv:math/0608781v2 [math.QA] 10 Apr 2007 ory, mechanical statistics, the theory of subfactors, and others, the common theme being “quantum symmetries”. A comprehensive presentation of fusion categories is [ENO], see also references therein. We refer to loc. cit. for definitions and notations used in the present paper. There is a notion of “module category over a tensor category” known in category theory since the sixties [Be]. Semisimple module categories over a fusion category should play the same fundamental rˆole as the representation theory of finite groups; see [O1, O2].
    [Show full text]
  • Self-Dual Modules for Finite Groups of Odd Order
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Algebra 330 (2011) 418–430 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Self-dual modules for finite groups of odd order ∗ Ping Jin a,b, ,1 a School of Mathematical Sciences, Shanxi University, Taiyuan 030006, PR China b School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China article info abstract Article history: Let G be a finite group of odd order and let F be a finite field. Received 28 February 2010 Suppose that V is an FG-module which carries a G-invariant non- Available online 21 December 2010 degenerate bilinear form which is symmetric or symplectic. We Communicated by Michel Broué show that V contains a self-perpendicular submodule if and only if the characteristic polynomials of some specified elements of G Keywords: Self-dual module (regarded as linear transformations of V ) are precisely squares. Symplectic form This result can be applied to the study of monomial characters if Symmetric form the form on V is symplectic, and self-dual group codes if the form Hyperbolic module is symmetric. Self-dual group code © 2010 Elsevier Inc. All rights reserved. 1. Introduction Let G be a finite group and let F be a finite field. Suppose that V is a finite-dimensional FG- module which carries a G-invariant non-degenerate bilinear form b.Ifb is symmetric then we call V a symmetric F G-module, and if b is symplectic, that is, if b is skew-symmetric and in addition when F has characteristic 2, b(v, v) = 0forallv in V , then we call V a symplectic F G-module.
    [Show full text]
  • Arxiv:1512.09344V1 [Math.RT] 31 Dec 2015 H S Ftecui Rpry Eepanhwsm Eut in Results Coalg Some for How Construction Explain Counitalization We a Using Property
    SEMIPERFECT AND COREFLEXIVE COALGEBRAS S.DASC˘ ALESCU˘ 1 AND M.C. IOVANOV1,2 Abstract. We study non-counital coalgebras and their dual non-unital algebras, and intro- duce the finite dual of a non-unital algebra. We show that a theory that parallels in good part the duality in the unital case can be constructed. Using this, we introduce a new notion of left coreflexivity for counital coalgebras, namely, a coalgebra is left coreflexive if C is isomorphic ∗ canonically to the finite dual of its left rational dual Rat(C∗ C ). We show that right semiper- fectness for coalgebras is in fact essentially equivalent to this left reflexivity condition, and we give the connection to usual coreflexivity. As application, we give a generalization of some recent results connecting dual objects such as quiver or incidence algebras and coalgebras, and show that Hopf algebras with non-zero integrals (compact quantum groups) are coreflexive. 1. Introduction and Preliminaries Let K be a field and let Γ be a quiver. Two combinatorial objects that are important in representation theory are associated with Γ: the quiver algebra K[Γ], and the path coalgebra KΓ. The quiver algebra is an algebra with enough idempotents, but does not have a unit unless Γ is finite, while the path coalgebra KΓ is a coalgebra with counit. In [5] it was investigated how one of the two objects can be recovered from the other one; a finite dual coalgebra A0 was constructed for an algebra A with enough idempotents, and it was proved that KΓ is isomorphic to K[Γ]0 provided that Γ has no oriented cycles and between any two vertices of Γ there are finitely many arrows.
    [Show full text]