DOCSLIB.ORG

Kac-Moody Algebras and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Kac-Moody Algebras and Applications Kac-Moody Algebras and Applications Owen Barrett December 24, 2014 Abstract This article is an introduction to the theory of Kac-Moody algebras: their genesis, their con- struction, basic theorems concerning them, and some of their applications. We first record some of the classical theory, since the Kac-Moody construction generalizes the theory of simple finite-dimensional Lie algebras in a closely analogous way. We then introduce the construction and properties of Kac-Moody algebras with an eye to drawing natural connec- tions to the classical theory. Last, we discuss some physical applications of Kac-Moody alge- bras,includingtheSugawaraandVirosorocosetconstructions,whicharebasictoconformal field theory. Contents 1 Introduction2 2 Finite-dimensional Lie algebras3 2.1 Nilpotency.....................................3 2.2 Solvability.....................................4 2.3 Semisimplicity...................................4 2.4 Root systems....................................5 2.4.1 Symmetries................................5 2.4.2 The Weyl group..............................6 2.4.3 Bases...................................6 2.4.4 The Cartan matrix.............................7 2.4.5 Irreducibility...............................7 2.4.6 Complex root systems...........................7 2.5 The structure of semisimple Lie algebras.....................7 2.5.1 Cartan subalgebras............................8 2.5.2 Decomposition of g ............................8 2.5.3 Existence and uniqueness.........................8 2.6 Linear representations of complex semisimple Lie algebras...........9 2.6.1 Weights and primitive elements.....................9 2.7 Irreducible modules with a highest weight....................9 2.8 Finite-dimensional modules............................ 10 3 Kac-Moody algebras 10 3.1 Basic definitions.................................. 10 3.1.1 Construction of the auxiliary Lie algebra................. 11 3.1.2 Construction of the Kac-Moody algebra................. 12 3.1.3 Root space of the Kac-Moody algebra g(A) ............... 12 1 3.2 Invariant bilinear form............................... 13 3.3 Casimir element.................................. 14 3.4 Integrable representations............................. 15 3.5 Weyl group..................................... 15 3.5.1 Weyl chambers.............................. 16 3.6 Classification of generalized Cartan matrices................... 16 3.6.1 Kac-Moody algebras of finite type.................... 17 3.7 Root system.................................... 17 3.7.1 Real roots................................. 17 3.7.2 Imaginary roots.............................. 18 3.7.3 Hyperbolic type.............................. 19 3.8 Kac-Moody algebras of affine type......................... 20 3.8.1 The underlying finite-dimensional Lie algebra.............. 20 3.8.2 Relating g and g˚ .............................. 21 4 Highest-weight modules over Kac-Moody algebras 22 4.1 Inductive Verma construction........................... 24 4.2 Primitive vectors and weights........................... 24 4.3 Complete reducibility............................... 25 4.4 Formal characters................................. 26 4.5 Reducibility..................................... 26 5 Physical applications of Kac-Moody algebras 28 5.1 Sugawara construction............................... 28 5.2 Virasoro algebra.................................. 30 5.3 Virasoro operators................................. 31 5.4 Coset Virasoro construction............................ 32 1 Introduction Élie Cartan, in his 1894 thesis, completed the classification of the finite-dimensional simple Lie algebras over C. Cartan's thesis is a convenient marker for the start of over a half-century of research on the structure and representation theory of finite-dimensional simple complex Lie algebras. This effort, marshalled into existence by Lie, Killing, Cartan, and Weyl, was contin- ued and extended by the work of Chevalley, Harish-Chandra, and Jacobson, and culminated in Serre's Theorem [Ser66], which gives a presentation for all finite-dimensional simple complex Lie algebras. In the mid-1960s, Robert Moody and Victor Kac independently introduced a collection of Lie algebras that did not belong to Cartan's classification. Moody, at the University of Toronto, con- structed Lie algebras from a generalization of the Cartan matrix, emphasizing the algebras that would come to be known as affine Kac-Moody Lie algebras, while Kac, at Moscow State Univer- sity, began by studying simple graded Lie algebras of finite growth. In this way, Moody and Kac independently introduced a new class of generally infinite-dimensional, almost always simple Lie algebras that would come to be known as Kac-Moody Lie algebras. The first major application of the infinite-dimensional Lie algebras of Kac and Moody came in 1972 with Macdonald's 1972 discovery [Mac71] of an affine analogue to Weyl's denominator formula. Specializing Macdonald's formula to particular affine root systems yielded previously- known identities for modular forms such as the Dedekind η-function and the Ramanujan τ- function. The arithmetic connection Macdonald established to affine Kac-Moody algebras was 2 subsequently clarified and explicated independently by both Kac and Moody, but only marked the beginning of the applications of these infinite-dimensional Lie algebras. It didn't take long for the importance of Kac-Moody algebras to be recognized, or for many applications of them to both pure mathematics and physics to be discovered. We will structure this article as follows. We will first recall some basic results in the theory of finite-dimensional Lie algebras so as to ease the introduction of their infinite-dimensional relatives. Next, we will define what is a Kac-Moody algebra, and present some fundamental theorems concerning these algebras. With the basic theory established, we will turn to some applications of Kac-Moody algebras to physics. We include proofs for none of the results quoted in this review, but we do take care to cite the references where appropriate proofs may be found. 2 Finite-dimensional Lie algebras We collect in this section some basic results pertaining to the structure and representation the- ory of finite-dimensional Lie algebras. This is the classical theory due to Lie, Killing, Cartan, Chevalley, Harish-Chandra, and Jacobson. The justification for reproducing the results in the finite-dimensional case is that both Moody and Kac worked initially to generalize the methods of Harish-Chandra and Chevalley as presented by Jacobson in [Jac62, Chapter 7]. For this reason, the general outline of the approach to studying the structure of Kac-Moody algebras resembles that of the approach to studying the finite-dimensional case. We follow the presentation of Serre [Ser00] in this section. We suppose throughout this sec- tion that g is a finite-dimensional Lie algebra over the field of complex numbers, though we will omit both ‘finite-dimensional’ and ‘complex’ in the statements that follow. The Lie bracket of x and y is denoted by [x; y] and the map y 7! [x; y] is denoted ad x. For basic definitions such as that of an ideal of a Lie algebra (and for proofs of some statements given without proof in [Ser00]), c.f. Humphreys [Hum72]. 2.1 Nilpotency n The lower central series of g is the descending series (C g)n>1 of ideals of g defined by the for- mulæ C1g = g (2.1) n n-1 C g = [g;C g] if n > 2. (2.2) Definition. A Lie algebra g is said to be nilpotent if there exists an integer n such that Cng = 0. g is nilpotent of class r if Cr+1g = 0. Proposition 2.1. The following conditions are equivalent. (i) g is nilpotent of class 6 r. (ii) There is a descending series of ideals g = a0 ⊃ a1 ⊃ · · · ⊃ ar = 0 (2.3) such that [g; ai] ⊂ ai+1 for 0 6 i 6 r - 1. Recall that the center of a Lie algebra is the set of x 2 g such that [x; y] = 0 for all y 2 g. It is an abelian ideal of g. We denote the center of g by Z(g). 3 Proposition 2.2. Let g be a Lie algebra and let a ⊂ Z(g) be an ideal. Then g is nilpotent , g=a is nilpotent. (2.4) We now present the basic results pertaining to nilpotent Lie algebras. Theorem 2.3 (Engel). For a Lie algebra g to be nilpotent, it is necessary and sufficient for ad x to be nilpotent 8x 2 g. Theorem 2.4. Let φ : g ! End(V) be a linear representation of a Lie algebra g on a nonzero finite- dimensionalvectorspaceV. Supposethatφ(x)isnilpotent8x 2 g. Then90 6= v 2 V whichisinvariant under g. 2.2 Solvability n The derived series of g is the descending series (D g)n>1 of ideals of g defined by the formulæ D1g = g (2.5) n n-1 n-1 D g = [D g;D g] if n > 2. (2.6) Definition. A Lie algebra g is said to be solvable if 9n 2 N s.t. Dng = 0. g is solvable of derived length r+1 6 r if D g = 0. n.b. Every nilpotent algebra is solvable. Proposition 2.5. The following conditions are equivalent. (i) g is solvable of derived length 6 r. (ii) There is a descending central series of ideals of g: g = a0 ⊃ a1 ⊃ · · · ⊃ ar = 0 (2.7) such that [ai; ai] ⊂ ai+1 for 0 6 i 6 r - 1 () successive quotients ai=ai+1 are abelian). In analogy with Engel's theorem, we now record Lie's theorem. Theorem 2.6 (Lie). Let φ : g ! End(V) be a finite-dimensional linear representation of g. If g is solv- able,thereisaflagDofV suchthatφ(g) ⊂ b(D),whereb(D)istheBorelalgebraofD(thesubalgebra of End V that stabilizes the flag). Theorem 2.7. If g is solvable, all simple g-modules are one-dimensional. Theorem 2.7 leads us to conclude that the representation theory of solvable Lie algebras is very boring. The following section will make this statement more transparent. 2.3 Semisimplicity Recall that the radical rad g is largest solvable ideal in g. rad g exists because if a and b are solv- able ideals of g, the ideal a + b is also solvable since it is an extension of b=(a \ b) = (a + b)=a by a. Theorem 2.8. Let g be a Lie algebra. (a) rad(g= rad g) = 0.
Load more

Recommended publications
  • 3D Gauge Theories, Symplectic Duality and Knot Homology I
    3d Gauge Theories, Symplectic Duality and Knot Homology I Tudor Dimofte Notes by Qiaochu Yuan December 8, 2014 We want to study some 3d N = 4 supersymmetric QFTs. They won't be topological, but the supersymmetry will still help. We'll look in particular at boundary conditions and compactifications to 2d. Most of this material is relatively old by physics standards (known since the '90s). A motivation for discussing these theories here is that they seem closely related to 1. geometric representation theory, 2. geometric constructions of knot homologies, and 3. symplectic duality. This story gives rise to some interesting mathematical predictions. On Tuesday we'll talk about what symplectic n-ality should mean. On Wednesday we'll give a new construction of projectives in variants of category O. On Thursday we'll talk about category O and knot homologies, and a new approach to both via 2d Landau-Ginzburg models. 1 Geometric representation theory For starters, let G = SL2. The universal enveloping algebra of its Lie algebra is U(sl2) = hE; F; H j [E; F ] = H; [H; E] = 2E; [H; F ] = −2F i: (1) The center is generated by the Casimir operator 1 χ = EF + FE + H2: (2) 2 The full category of U(sl2)-modules is hard to work with. Bernstein, Gelfand, and Gelfand proposed that we work in a nice subcategory O of highest weight representations (on which E acts locally finitely or locally nilpotently, depending on the definition) decomposing as a sum M M = Mµ (3) µ of generalized eigenspaces with respect to the action of the generator of the Cartan L subalgebra H.
    [Show full text]
  • Parabolic Category O for Classical Lie Superalgebras
    Parabolic category O for classical Lie superalgebras Volodymyr Mazorchuk Abstract We compare properties of (the parabolic version of) the BGG category O for semi-simple Lie algebras with those for classical (not necessarily simple) Lie superalgebras. 1 Introduction Category O for semi-simple complex finite dimensional Lie algebras, introduced in [BGG], is a central object of study in the modern representation theory (see [Hu]) with many interesting connections to, in particular, combinatorics, alge- braic geometry and topology. This category has a natural counterpart in the super- world and this super version O˜ of category O was intensively studied (mostly for some particular simple classical Lie superalgebra) in the last decade, see e.g. [Br1, Br2, Go2, FM2, CLW, CMW] or the recent books [Mu, CW] for details. The aim of the present paper is to compare some basic but general properties of the category O in the non-super and super cases for a rather general classical super-setup. We mainly restrict to the properties for which the non-super and super cases can be connected using the usual restriction and induction functors and the biadjunction (up to parity change) between these two functors is our main tool. The paper also complements, extends and gives a more detailed exposition for some results which appeared in [AM, Section 7]. The original category O has a natural parabolic version which first appeared in [RC]. We start in Section 2 by setting up an elementary approach (using root system geometry) to the definition of a parabolic category O for classical finite dimensional Lie superalgebras.
    [Show full text]
  • Geometric Representation Theory, Fall 2005
    GEOMETRIC REPRESENTATION THEORY, FALL 2005 Some references 1) J. -P. Serre, Complex semi-simple Lie algebras. 2) T. Springer, Linear algebraic groups. 3) A course on D-modules by J. Bernstein, availiable at www.math.uchicago.edu/∼arinkin/langlands. 4) J. Dixmier, Enveloping algebras. 1. Basics of the category O 1.1. Refresher on semi-simple Lie algebras. In this course we will work with an alge- braically closed ground field of characteristic 0, which may as well be assumed equal to C Let g be a semi-simple Lie algebra. We will fix a Borel subalgebra b ⊂ g (also sometimes denoted b+) and an opposite Borel subalgebra b−. The intersection b+ ∩ b− is a Cartan subalgebra, denoted h. We will denote by n, and n− the unipotent radicals of b and b−, respectively. We have n = [b, b], and h ' b/n. (I.e., h is better to think of as a quotient of b, rather than a subalgebra.) The eigenvalues of h acting on n are by definition the positive roots of g; this set will be denoted by ∆+. We will denote by Q+ the sub-semigroup of h∗ equal to the positive span of ∆+ (i.e., Q+ is the set of eigenvalues of h under the adjoint action on U(n)). For λ, µ ∈ h∗ we shall say that λ ≥ µ if λ − µ ∈ Q+. We denote by P + the sub-semigroup of dominant integral weights, i.e., those λ that satisfy hλ, αˇi ∈ Z+ for all α ∈ ∆+. + For α ∈ ∆ , we will denote by nα the corresponding eigen-space.
    [Show full text]
  • Arxiv:1907.06685V1 [Math.RT]
    CATEGORY O FOR TAKIFF sl2 VOLODYMYR MAZORCHUK AND CHRISTOFFER SODERBERG¨ Abstract. We investigate various ways to define an analogue of BGG category O for the non-semi-simple Takiff extension of the Lie algebra sl2. We describe Gabriel quivers for blocks of these analogues of category O and prove extension fullness of one of them in the category of all modules. 1. Introduction and description of the results The celebrated BGG category O, introduced in [BGG], is originally associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra. The definition of O is naturally generalized to all Lie algebras admitting some analogue of a triangular decomposition, see [MP]. These include, in particular, Kac-Moody algebras and Virasoro algebra. Category O has a number of spectacular properties and applications to various areas of mathematics, see for example [Hu] and references therein. The paper [DLMZ] took some first steps in trying to understand structure and prop- erties of an analogue of category O in the case of a non-reductive finite dimensional Lie algebra. The investigation in [DLMZ] focuses on category O for the so-called Schr¨odinger algebra, which is a central extension of the semi-direct product of sl2 with its natural 2-dimensional module. It turned out that, for the Schr¨odinger algebra, the behavior of blocks of category O with non-zero central charge is exactly the same as the behavior of blocks of category O for the algebra sl2. At the same, block with zero central charge turned out to be significantly more difficult.
    [Show full text]
  • Quantizations of Regular Functions on Nilpotent Orbits 3
    QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS IVAN LOSEV Abstract. We study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa. 1. Introduction 1.1. Nilpotent orbits and their quantizations. Let G be a connected semisimple algebraic group over C and let g be its Lie algebra. Pick a nilpotent orbit O ⊂ g. This orbit is a symplectic algebraic variety with respect to the Kirillov-Kostant form. So the algebra C[O] of regular functions on O acquires a Poisson bracket. This algebra is also naturally graded and the Poisson bracket has degree −1. So one can ask about quantizations of O, i.e., filtered algebras A equipped with an isomorphism gr A −→∼ C[O] of graded Poisson algebras. We are actually interested in quantizations that have some additional structures mir- roring those of O. Namely, the group G acts on O and the inclusion O ֒→ g is a moment map for this action. We want the G-action on C[O] to lift to a filtration preserving action on A.
    [Show full text]
  • Classifying the Representation Type of Infinitesimal Blocks of Category
    Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt (Under the Direction of Brian D. Boe) Abstract Let g be a simple Lie algebra over the field C of complex numbers, with root system Φ relative to a fixed maximal toral subalgebra h. Let S be a subset of the simple roots ∆ of Φ, which determines a standard parabolic subalgebra of g. Fix an integral weight ∗ µ µ ∈ h , with singular set J ⊆ ∆. We determine when an infinitesimal block O(g,S,J) := OS of parabolic category OS is nonzero using the theory of nilpotent orbits. We extend work of Futorny-Nakano-Pollack, Br¨ustle-K¨onig-Mazorchuk, and Boe-Nakano toward classifying the representation type of the nonzero infinitesimal blocks of category OS by considering arbitrary sets S and J, and observe a strong connection between the theory of nilpotent orbits and the representation type of the infinitesimal blocks. We classify certain infinitesimal blocks of category OS including all the semisimple infinitesimal blocks in type An, and all of the infinitesimal blocks for F4 and G2. Index words: Category O; Representation type; Verma modules Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt B.A., Utah State University, 1999 M.S., Utah State University, 2001 M.A, The University of Georgia, 2006 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2008 c 2008 Kenyon J. Platt All Rights Reserved Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J.
    [Show full text]
  • On the Left Ideal in the Universal Enveloping
    proceedings of the american mathematical society Volume 121, Number 4, August 1994 ON THE LEFT IDEAL IN THE UNIVERSALENVELOPING ALGEBRA OF A LIE GROUP GENERATED BY A COMPLEX LIE SUBALGEBRA JUAN TIRAO (Communicated by Roe W. Goodman) Abstract. Let Go be a connected Lie group with Lie algebra go and 'et n be a Lie subalgebra of the complexification g of go . Let C°°(Go)* be the annihilator of h in C°°(Go) and let s/ =^{Ccc(G0)h) be the annihilator of C00(Go)A in the universal enveloping algebra %(g) of g. If h is the complexification of the Lie algebra ho of a Lie subgroup Ho of Go then sf = %({g)h whenever Ho is closed, is a known result, and the point of this paper is to prove the converse assertion. The paper has two distinct parts, one for C°° , the other for holomorphic functions. In the first part the Lie algebra ho of the closure of Ha is characterized as the annihilator in go of C°°(Go)h , and it is proved that ho is an ideal in /i0 and that ho = ho © v where v is an abelian subalgebra of Ao . In the second part we consider a complexification G of Go and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that s/{cf(G)h) = %?(g)h if and only if G/H has many holomorphic functions. This is the case if G/H is a quasi-affine variety. From this we get that if H is a unipotent subgroup of G or if G and H are reductive groups then s/(C°°(G0)A) = iY(g)h .
    [Show full text]
  • Lie Groups and Lie Algebras
    2 LIE GROUPS AND LIE ALGEBRAS 2.1 Lie groups The most general definition of a Lie group G is, that it is a differentiable manifold with group structure, such that the multiplication map G G G, (g,h) gh, and the inversion map G G, g g−1, are differentiable.× → But we shall7→ not need this concept in full generality.→ 7→ In order to avoid more elaborate differential geometry, we will restrict attention to matrix groups. Consider the set of all invertible n n matrices with entries in F, where F stands either for the real (R) or complex× (C) numbers. It is easily verified to form a group which we denote by GL(n, F), called the general linear group in n dimensions over the number field F. The space of all n n matrices, including non- invertible ones, with entries in F is denoted by M(n,×F). It is an n2-dimensional 2 vector space over F, isomorphic to Fn . The determinant is a continuous function det : M(n, F) F and GL(n, F) = det−1(F 0 ), since a matrix is invertible → −{ } 2 iff its determinant is non zero. Hence GL(n, F) is an open subset of Fn . Group multiplication and inversion are just given by the corresponding familiar matrix 2 2 2 2 2 operations, which are differentiable functions Fn Fn Fn and Fn Fn respectively. × → → 2.1.1 Examples of Lie groups GL(n, F) is our main example of a (matrix) Lie group. Any Lie group that we encounter will be a subgroups of some GL(n, F).
    [Show full text]
  • Introduction to Representations Theory of Lie Groups
    Introduction to Representations Theory of Lie Groups Raul Gomez October 14, 2009 Introduction The purpose of this notes is twofold. The first goal is to give a quick answer to the question \What is representation theory about?" To answer this, we will show by examples what are the most important results of this theory, and the problems that it is trying to solve. To make the answer short we will not develop all the formal details of the theory and we will give preference to examples over proofs. Few results will be proved, and in the ones were a proof is given, we will skip the technical details. We hope that the examples and arguments presented here will be enough to give the reader and intuitive but concise idea of the covered material. The second goal of the notes is to be a guide to the reader interested in starting a serious study of representation theory. Sometimes, when starting the study of a new subject, it's hard to understand the underlying motivation of all the abstract definitions and technical lemmas. It's also hard to know what is the ultimate goal of the subject and to identify the important results in the sea of technical lemmas. We hope that after reading this notes, the interested reader could start a serious study of representation theory with a clear idea of its goals and philosophy. Lets talk now about the material covered on this notes. In the first section we will state the celebrated Peter-Weyl theorem, which can be considered as a generalization of the theory of Fourier analysis on the circle S1.
    [Show full text]
  • Scientific Report for the Year 2000
    The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Scientific Report for the Year 2000 Vienna, ESI-Report 2000 March 1, 2001 Supported by Federal Ministry of Education, Science, and Culture, Austria ESI–Report 2000 ERWIN SCHRODINGER¨ INTERNATIONAL INSTITUTE OF MATHEMATICAL PHYSICS, SCIENTIFIC REPORT FOR THE YEAR 2000 ESI, Boltzmanngasse 9, A-1090 Wien, Austria March 1, 2001 Honorary President: Walter Thirring, Tel. +43-1-4277-51516. President: Jakob Yngvason: +43-1-4277-51506. [email protected] Director: Peter W. Michor: +43-1-3172047-16. [email protected] Director: Klaus Schmidt: +43-1-3172047-14. [email protected] Administration: Ulrike Fischer, Eva Kissler, Ursula Sagmeister: +43-1-3172047-12, [email protected] Computer group: Andreas Cap, Gerald Teschl, Hermann Schichl. International Scientific Advisory board: Jean-Pierre Bourguignon (IHES), Giovanni Gallavotti (Roma), Krzysztof Gawedzki (IHES), Vaughan F.R. Jones (Berkeley), Viktor Kac (MIT), Elliott Lieb (Princeton), Harald Grosse (Vienna), Harald Niederreiter (Vienna), ESI preprints are available via ‘anonymous ftp’ or ‘gopher’: FTP.ESI.AC.AT and via the URL: http://www.esi.ac.at. Table of contents General remarks . 2 Winter School in Geometry and Physics . 2 Wolfgang Pauli und die Physik des 20. Jahrhunderts . 3 Summer Session Seminar Sophus Lie . 3 PROGRAMS IN 2000 . 4 Duality, String Theory, and M-theory . 4 Confinement . 5 Representation theory . 7 Algebraic Groups, Invariant Theory, and Applications . 7 Quantum Measurement and Information . 9 CONTINUATION OF PROGRAMS FROM 1999 and earlier . 10 List of Preprints in 2000 . 13 List of seminars and colloquia outside of conferences .
    [Show full text]
  • Lie Groups and Lie Algebras
    Lie groups and Lie algebras Jonny Evans March 10, 2016 1 Lecture 1: Introduction To the students, past, present and future, who have/are/will taken/taking/take this course and to those interested parties who just read the notes and gave me feedback: thank you for providing the background level of enthusiasm, pedantry and general con- fusion needed to force me to improve these notes. All comments and corrections are very welcome. 1.1 What is a representation? Definition 1.1. A representation ρ of a group G on an n-dimensional vector space V over a field k is an assignment of a k-linear map ρ(g): V ! V to each group element g such that • ρ(gh) = ρ(g)ρ(h) • ρ(1G) = 1V . Here are some equivalent definitions. A representation is... 1. ...an assignment of an n-by-n matrix ρ(g) to each group element g such that ρ(gh) = ρ(g)ρ(h) (i.e. the matrices multiply together like the group elements they represent). This is clearly the same as the previous definition once you have picked a basis of V to identify linear maps with matrices; 2. ...a homomorphism ρ: G ! GL(V ). The multiplicativity condition is now hiding in the definition of homomorphism, which requires ρ(gh) = ρ(g)ρ(h); 3. ...an action of G on V by linear maps. In other words an action ρ~: G × V ! V such that, for each g 2 G, the map ρ(g): V ! V defined by ρ(g)(v) =ρ ~(g; v) is linear.
    [Show full text]
  • Grothendieck Rings of Basic Classical Lie Superalgebras
    Annals of Mathematics 173 (2011), 663{703 doi: 10.4007/annals.2011.173.2.2 Grothendieck rings of basic classical Lie superalgebras By Alexander N. Sergeev and Alexander P. Veselov Abstract The Grothendieck rings of finite dimensional representations of the basic classical Lie superalgebras are explicitly described in terms of the corre- sponding generalized root systems. We show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super Weyl groupoids. Contents 1. Introduction 663 2. Grothendieck rings of Lie superalgebras 665 3. Basic classical Lie superalgebras and generalized root systems 667 4. Ring J(g) and supercharacters of g 668 5. Geometry of the highest weight set 670 6. Proof of the main theorem 677 7. Explicit description of the rings J(g) 681 8. Special case A(1; 1) 695 9. Super Weyl groupoid 698 10. Concluding remarks 700 References 701 1. Introduction The classification of finite-dimensional representations of the semisimple complex Lie algebras and related Lie groups is one of the most beautiful pieces of mathematics. In his essay [1] Michael Atiyah mentioned representation the- ory of Lie groups as an example of a theory, which \can be admired because of its importance, the breadth of its applications, as well as its rational coher- ence and naturality." This classical theory goes back to fundamental work by Elie` Cartan and Hermann Weyl and is very well presented in various books, of 663 664 ALEXANDER N. SERGEEV and ALEXANDER P. VESELOV which we would like to mention the famous Serre's lectures [27] and a nicely written Fulton-Harris course [9].
    [Show full text]