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A Thesis Entitled on Weight Modules Over Sl2 by Hanh Nguyen

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A Thesis Entitled on Weight Modules Over Sl2 by Hanh Nguyen A Thesis entitled On weight modules over sl2 by Hanh Nguyen Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Arts in Mathematics Dr. Akaki Tikaradze, Committee Chair Dr. Gerard Thompson, Committee Member Dr. Mao-Pei Tsui, Committee Member Dr. Patricia R. Komuniecki, Dean College of Graduate Studies The University of Toledo August 2013 Copyright 2013, Hanh Nguyen This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of On weight modules over sl2 by Hanh Nguyen Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Master of Arts in Mathematics The University of Toledo August 2013 An important class of representations of sl2 consists of weight modules. The goal of this thesis is to give a self-contained exposition of the classification of simple weight modules over C. iii Acknowledgments I would like to express my deepest gratitude to my advisor Professor Akaki Tikaradze. Without his guidance and persistent help, this thesis would not have been possible. I would also like to offer special thanks to my loving family, especially my husband Trieu Le for his encouragement and help. Last but not least, I would like to thank my professors and friends for all the support. iv Contents Abstract iii Acknowledgments iv Contents v 1 Basic Definitions and Introduction 1 2 Universal Enveloping Algebras 9 3 Verma modules and their quotients 11 3.1 sl2-modules which have a maximal or a minimal vector . 13 3.2 Complete Reducibility . 15 4 sl2 -modules which have neither a maximal or a minimal vector 19 References 24 v Chapter 1 Basic Definitions and Introduction We start by recalling few basic notions related to Lie algebras. Definition 1.1. A vector space g over C, with an operation g × g ! g denoted (x; y) 7! [x; y] and called the bracket or commutator of x and y, is called a Lie algebra over C if the following satisfied: (L1) The bracket operation is bilinear. (L2)[ x; x] = 0 for all x in g (L3)[ x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0 for all x; y; z in g Remark 1.2. (L3) is called the Jacobi identity. Remark 1.3. (L3) shows that g is not associative. Remark 1.4. Applying (L1) and (L2), we obtain 0 = [x + y; x + y] = [x; x + y] + [y; x + y] = [x; x] + [x; y] + [y; x] + [y; y] = [x; y] + [y; x]: It follows that [x; y] = −[y; x] for all x; y in g. Hence [·; ·] is skew symmetric. 1 Example 1.5. Any vector space V over any field is a Lie algebra with the trivial bracket operation: [x; y] = 0 for all x; y in V . Example 1.6. General linear algebra. Let V be a vector space over C. Let End(V ) denote the set of all linear transformations from V into itself. Then End(V ) is an associative ring with respect to the usual composition operation of linear trans- formations. Now we define a new operation [x; y] = xy − yx, then End(V ) becomes a Lie algebra over C. We write gl(V ) for End(V ) and call it the general linear algebra. Below we check that the bracket [x; y] satisfies all three axioms (L1),(L2) and (L3) in the definition. (L1) Since the composition of linear maps is bilinear, this axiom is satisfied. (L2) For all x in gl(V ), we have [x; x] = xx − xx = 0. (L3) For all x; y; z in gl(V ), we compute [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = [x; yz − zy] + [y; zx − zx] + [z; xy − yx] = x(yz − zy) − (yz − zy)x + y(zx − xz) − (zx − xz)y + z(xy − yx) − (xy − yx)z = xyz − xzy − yzx + zyx + yzx − yxz − zxy + xzy + zxy − zyx − xyz + yxz = 0: Thus, axiom (L3) is also satisfied. Definition 1.7. A subspace K is a subalgebra of g if and only if K is closed under the bracket operation, that is, for every x; y 2 K,[x; y] 2 K. Example 1.8. Let V be a finite dimensional vector space. Let sl(V ) be the set of all linear transformations on V with trace zero. Then sl(V ) is in fact a subalgebra 2 of gl(V ). First, using the identity Tr(x + y) = Tr(x) + Tr(y) which holds for all x; y 2 gl(V ), we infer that sl(V ) is a linear subspace of gl(V ). Second, it follows from linearity of the trace and the identity Tr(xy) = Tr(yx) that Tr([x; y]) = Tr(xy − yx) = Tr(xy) − Tr(yx) = 0: Thus, [x; y] 2 sl(V ), which shows that sl(V ) is closed under the bracket operation. We call sl(V ) the special linear algebra over V . We now discuss the dimension of sl(V ) in the case V has finite dimension. Write dim(V ) = n + 1. Because sl(V ) is a proper subspace of gl(V ) (note that the identity transformation does not belong to sl(V )) and dim(gl(V )) = (n + 1)2, we see that 2 dim(sl(V )) 6 (n + 1) − 1. On the other hand, we may exhibit this number of linearly independent matrices of trace zero by taking all ei;j (i 6= j) along with hi = ei;i − ei+1;i+1(1 6 i 6 n). It can be checked that these matrices are linearly 2 independent. We have (n + 1) − (n + 1) such ei;j matrices and n such hi matrices. This shows that the number of matrices we have is (n+1)2 −(n+1)+n = (n+1)2 −1. We conclude that dim(sl(V )) = (n + 1)2 − 1. 2 For an example, consider the case when V = C . We use sl2 to denote sl(V ). With the above notation, we have 0 1 0 1 0 1 0 1 0 0 1 0 B C B C B C e1;2 = @ A ; e2;1 = @ A ; h1 = e1;1 − e2;2 = @ A : 0 0 1 0 0 −1 Put e := e1;2, f := e2;1, h := h1. Then it follows from the above discussion that the collection fe; f; hg is a basis of sl(2; C). Direct calculations show [e; f] = h; [h; e] = 2e; [h; f] = 2f: 3 Definition 1.9. Let g and g0 be Lie algebra over C. A linear transformation φ : g ! g0 is called a homomorphism between g and g0 if φ([x; y]) = [φ(x); φ(y)] for all x; y in g. Definition 1.10. Let g be a Lie algebra. A representation of g is a homomorphism φ : g ! gl(V ), where V is a vector space over C. Here V can be finite or infinite dimensional. Definition 1.11. A vector space V over C, endowed with an operation g × V ! V (denoted (x; v) 7! x · v or just xv) is called a g-module if for all x; y 2 g, all v; w 2 V , and a; b 2 F , the following conditions are satisfied: (M1)( ax + by) · v = a(x · v) + b(y · v), (M2) x · (av + bw) = a(x · v) + b(x · w), (M3)[ x; y] · v = x · (y · v) − y · (x · v). Clearly above two notions are the same. It will be convenient to use the language of representations along with the language of modules. Example 1.12. Let g be a Lie algebra over C. Define Der(g) to be the set of all linear maps D : g ! g which satisfy the Leibniz rule D([x; y]) = [D(x); y] + [x; D(y)] for all x; y 2 g: Then Der(g) is a subspace of gl(g). Each linear map in Der(g) is called a derivation. We now define a special type of derivations. Each x 2 g defines a map adx : g ! g by adx(y) = [x; y] for all y 2 g. We claim that adx 2 Der(g). Indeed, adx is linear. For any α; β 2 F and y; z 2 g, we have adx(αy + βz) = [x; αy + βz] = α[x; y] + β[x; z] = α adx(y) + β adx(z): 4 Now we show that adx satisfies the Leibniz rule. For all y; z 2 g, adx([y; z]) = [x; [y; z]] = −[z; [x; y]] − [y; [z; x]] (from Jacobi identity) = [[x; y]; z] + [y; [x; z]] = [adx(y); z] + [y; adx(z)]: Consider the map ad defined by ad : g −! gl(g) x 7! adx We claim that ad is a homomorphism of Lie algebras. Indeed, ad is linear by the linearity of the Lie bracket. We now show that ad preserves the Lie bracket, that is, ad[x;y] = [adx; ady] = adx ◦ ady − ady ◦ adx for all x; y 2 g. For each z 2 g, ad[x;y](z) = [[x; y]; z] = −[z; [x; y] = [x; [y; z]] + [y; [z; x]] (by Jacobi identity) = [x; [y; z]] − [y; [x; z]] = adx([y; z]) − ady([x; z]) = adx(ady(z)) − ady(adx(z)) = [adx; ady](z): Thus, (g; ad) is a representation of g. Definition 1.13. Let V; W be g-modules. A homomorphism of g-modules is a linear map φ : V ! W such that φ(x · v) = x · φ(v) for all x 2 F and v 2 V . When φ is a vector space isomorphism then we call it an isomorphism of g-modules. Definition 1.14.
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