U.U.D.M. Project Report 2019:18 Verma Modules, The Weyl Character Formula and Embedding Theorems Aban Zehra Husain Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Juni 2019 Department of Mathematics Uppsala University Uppsala University Department of Mathematics Masters Thesis Verma Modules, The Weyl Character Formula and Embedding Theorems Supervisor: Author: Prof. Dr. Volodymyr Aban Zehra Husain Mazorchuk 2019 Contents 1 Preliminaries 1 1.1 Lie Algebras . 1 1.2 Modules . 3 1.3 The Cartan Subalgebra and the Root System . 4 1.4 The Universal Enveloping Algebra and the Poincaré Birkhoff Witt Theorem . 6 1.5 Complete Reducibility Theorem . 8 1.6 Weight and weight modules . 10 1.7 Induced Representations . 11 2 The Weyl Character Formula 11 2.1 Verma Modules . 11 2.2 The Weyl Character Formula . 15 2.2.1 The Weyl Dimension Formula . 17 2.3 Examples . 18 2.3.1 g = sl2 ...................................... 18 2.3.2 g = sl3 ...................................... 19 2.3.3 The Natural Representation . 19 2.3.4 The Adjoint Representation . 20 3 Homomorphisms Between Verma Modules 20 3.1 Submodules and Subquotients of Verma Modules . 21 3.2 Homomorphisms Between Verma Modules . 22 3.3 Existence of Embeddings (Integral Case) . 23 3.4 Existence of Embeddings (General Case) . 25 3.5 Antidominant Weights and Simple Verma Modules . 25 3.6 The example of M(ρ) ................................. 26 3.6.1 The chains corresponding to α and β ..................... 26 3.6.2 The chains containing both α and β ..................... 27 Aban Husain 1 1 Preliminaries Unless otherwise specified, all vector spaces are over C. 1.1 Lie Algebras Definition 1.1. A vector space g equipped with a bilinear operation [·; ·] (sometimes called a Lie bracket) such that: (i) [x; x] = 0 for all x 2 g; (ii) [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0 for all x; y; z 2 g: is a Lie algebra. If A is an associative algebra over a field, a Lie bracket may be defined by the commutator [x; y] = xy − yx for all x; y 2 A, where (x; y) 7! xy is the multiplicative operation on A. Then (A; [·; ·]) is called the underlying Lie Algebra of A, and A is referred to as an enveloping algebra. Every Lie algebra may be embedded into the underlying Lie algebra for some associative algebra. Definition 1.2. A vector subspace g0 of g is a Lie subalgebra if [x; y] 2 g0 for all x; y 2 g0: Definition 1.3. An ideal is a subalgebra g0 of g such that [x; y] 2 g0 for all x 2 g and y 2 g0: If g0 is an ideal, the bracket operation defines a bracket operation on g=g0, so g=g0 is a Lie algebra, called the quotient Lie algebra. If g contains no proper nonzero ideals it is termed a simple Lie algebra and called a semi-simple Lie algebra if g decomposes into simple components. Let V be a vector space and End(V ) be the algebra of endomorphisms of V . The underlying Lie algebra of End(V ) is called gl(V ): Let V be finite dimensional with dimension n. Given a fixed basis of V , gl(V ) may be identified with the algebra of n × n matrices and denoted gln. The canonical basis of gln in given by eij (the n × n matrix with 1 in the (ij) position and 0 elsewhere). The subalgebras of gl(V ) are referred to as the linear Lie algebras. Example. Let V be a vector space of dimension n. The special linear algebra is the subalge- bra of gln consisting of endomorphisms with trace 0, denoted by sln: As trace is independent of basis, this subalgebra is well defined. Given that Tr(xy) = Tr(yx) and Tr(x+y) = Tr(x)+Tr(y); clearly sln is a Lie subalgebra of gln. A basis of sln may be formed by matrices of the form eij 2 for i 6= j and eii − ei+1;i+1 for 1 ≤ i < n, thus having dimension n − 1. Example. Let V be a vector space of odd dimension n = 2k + 1 with a bilinear form on V defined by the matrix 21 0 0 3 s = 40 0 Ik5 0 Ik 0 where Ik is the k × k identity matrix. The special orthogonal algebra is the subalgebra of gln t consisting of endomorphisms x such that sx = −x s, and is denoted son. If x is partitioned as s is, then an equivalent condition is that x must be of the form 2 3 0 b1 b2 t x = 4−b2 m n 5 t t −b1 p −m t for arbitrary 1 × k matrices b1; b2 and k × k matriced m; n; p with the added constraints n = −n and pt = −p: A basis may be constructed by restricting to each of the associated partitions. The matrices e1;k+i+1 − ei+1;1 and e1;i+1 − ek+i+1;1 for 1 ≤ i ≤ k form b2 and b1. Corresponding to m; n and p respectively, the matrices ei+1;j+1 − ek+i+1;k+j+1 for 1 ≤ i 6= j ≤ k, ei+1;k+j+1 − ej+1;k+i+1 for 1 ≤ i < j ≤ k and ek+i+1;j+1 − ek+j+1;i+1 for 1 ≤ i < j ≤ k form a partial basis. With the addition of diagonal matrices of the form eii − ek+i;k+i for 2 ≤ i ≤ k + 1 yields a basis 2 of son with dimension 2k + k: Aban Husain 2 In the case of an even dimensional vector space, the orthogonal algebra is constructed similarly, where the bilinear form on V is of the form 0 I s = k : Ik 0 Let g be a Lie algebra, and define an operation (x; y) 7! −[x; y]. Then g equipped with this new bracket operation becomes another Lie algebra called the opposite to g: Definition 1.4. Let g; g0 be Lie algebras. A linear map φ : g ! g0 is a homomorphism of Lie algebras if φ([x; y]) = [φ(x); φ(y)] for all x; y 2 g: Definition 1.5. Let X; Y be subsets of g. The set of elements of X that commute, i.e. the set of elements x such that [x; y] = 0, with all elements y of Y is called the centralizer of Y in X: The centralizer of g in g is an ideal and referred to as the center of g: Definition 1.6. The normalizer n of a subalgebra h in g is the subalgebra consisting of all x 2 g such that [x; h] 2 h: The subalgebra h is an ideal of n: Definition 1.7. Let g be a Lie algebra. The descending central series of g is the chain of subalgebras: g = g1 ⊇ g2 = [g; g1] ⊇ g3 = [g; g2] ⊇ · · · ⊇ gn = [g; gn−1] ⊇ :::; and the derived series is g = g(0) ⊇ g(1) = [g(0); g(0)] ⊇ g(2) = [g(1); g(1)] ⊇ · · · ⊇ g(n) = [g(n−1); g(n−1)] ⊇ :::: For any semisimple Lie algebra g; the derived subalgebra [g; g] is nonzero, and hence [g; g] = g given that g is a sum of simple Lie algebras. Definition 1.8. A Lie algebra g is nilpotent if it satisfies one of the following equivalent conditions: k (i) there exists k 2 Z≥0 such that g = 0; (ii) there exists k 2 Z≥0 such that [x1; [x2; [:::; [xk−1; xk];:::; ]]] = 0 for all x1; : : : ; xk 2 g; (iii) there exists a decreasing series of ideals of g : g1 ⊇ g2 ⊇ · · · ⊇ gn such that g1 = g, gn = 0 and [g; gi] ⊆ gi+1 for i < n. Definition 1.9. A Lie algebra g is solvable if it satisfies one of the following equivalent condi- tions: (k) (i) there exists k 2 Z≥0 such that g = 0; (ii) there exists a decreasing series of ideals of g : g1 ⊇ g2 ⊇ · · · ⊇ gn such that g1 = g, gn = 0 and [gi; gi] ⊆ gi+1 for i < n. By definition of the descending central series and derived series, g(k) ⊂ gn+1: Hence if the descending central series terminates, so must the derived series, i.e. every nilpotent subalgebra is solvable. Example. The subalgebra of gln consisting of strictly upper triangular matrices is nilpotent, and thus solvable. Conversely, the subalgebra consisting of upper triangular matrices is solvable only. Aban Husain 3 1.2 Modules Definition 1.10. Let V be a vector space. A homomorphism % : g ! gl(V ) is termed a representation of g in V . The dimension of the representation % is the dimension of the space V , and V is referred to as a g-module. The labels representation and module will be used interchangably. For simplicity, %(x)v is denoted xv unless specification is necessary. Example. Let g be any Lie algebra. Then g functions as a g-module via the action %(x)y = [x; y]: This is termed the adjoint representation. The endomorphism %(x) is denoted ad x. n Example. Given g = sln, the natural representation is the representation of sln in C such that each x 2 sln acts by normal matrix multiplication on vectors. This representation holds for any other Lie subalgebra of gln. Definition 1.11. Given representations %; %0 of g in V; V 0; a linear map φ : V ! V 0 is a g-homomorphism if %0(x)φ(v) = φ(%(x)v) for all x 2 g. Definition 1.12. Two representations %; %0 of g in V; V 0 are equivalent or isomorphic if there exists a g-isomorphism from V to V 0.