# Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers

Size: px
Start display at page:

Download "Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers"

Transcription

1 Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009

2 2

3 Contents 1 Background Linear algebra Bilinear forms Jordan decomposition Some key facts about Lie algebras Cartan-Killing form and semisimplicity Jordan decomposition Jacobson-Morozov Theorem sl 2 (C) Representations of sl 2 (C) The irreducible representation of dimension n Jacobson-Morozov Theorem Classifying nilpotent orbits Dynkin-Kostant classification Bala-Carter Theorem Classical groups Parametrizing orbits by partitions Partial order on orbits

4 4 CONTENTS

5 Chapter 1 Background In Chapter 1 we gather background material on linear algebra and Lie algebras that may get used in the lectures. 1.1 Linear algebra The theory of Lie algebras is heavily based on a few core results in linear algebra. This section details some of those results. Throughout we assume that vector spaces are over the complex numbers C Bilinear forms This subsection should be skipped for now and used as a reference. The Cartan-Killing form on a Lie algebra plays an important technical role. It is a symmetric, bilinear form on the Lie algebra (viewed as a vector space). Let V denote a vector space over C. A symmetric, bilinear form is a map V V C, written as (v, w) for v, w V. Bilinear means it is linear in each factor and symmetric means that (v, w) = (w, v). Once we choose a basis for V, any bilinear form can be expressed in terms of matrix multiplication as (v, w) = v T Aw, where v and w are written as column vectors and where A is an n n matrix with n = dim(v ). The bilinear form is symmetric if and only if A is a symmetric matrix. Usually we only care about bilinear forms up to a change of basis (much as happens for linear transformations). This amounts to classifying symmetric matrices up to transformation of A into B T AB, where B is an invertible matrix (why?). Now any symmetric matrix can be conjugated to a diagonal matrix by an orthogonal matrix (why?) and recall that orthogonal matrices B satisfy B 1 = B T. Hence any symmetric, bilinear forms can be written using a diagonal matrix after choosing the basis for V appropriately. Next, considering B T AB where A and B are diagonal, we can remove any perfect square divisor of the entries of A. In other words, over C the matrix A is equivalent to a diagonal matrix where every entry is 0 or 1. 5

6 6 CHAPTER 1. BACKGROUND Exercise Show that the number of zeros and ones obtained from this process is determines and is uniquely determined by A. In other words, the number of inequivalent symmetric bilinear forms over C is equal to n + 1, where n is the dimension of V. 2. What is the situation for classifying symmetric, bilinear forms over the real numbers R? (this is Sylvester s Theorem) Often we are interested in non-degenerate forms. This means that if (v, w) = 0 for all w V, then v = 0. Note that this implies the matrix A above is equivalent to the identity matrix, i.e. there are no zeros. Note that over C, there are vectors which are orthogonal to themselves, once the vector space has dimension at least two. For example, taking the usual form on C 2, then v = (1, i) satisfies (v, v) = 0. On the other hand, it is still the case that the orthogonal space to a subspace still has the expected dimension. Namely, Proposition Let V be vector space of dimension n with a non-degenerate, symmetric, bilinear form. Let U V be a subspace of dimension m. Define the orthogonal space Then U has dimension n m. U := {v V (v, u) = 0 for all u U}. Proof. For each u U, consider the linear map T u : V C given by T u (v) := (v, u). These are just elements of the dual vector space V of linear maps from V to C. Notice that U is in the kernel of each T u, so we can in fact consider T u as an element in (V/U ). Since the form is non-degenerate, if u is nonzero, then T u cannot be the zero element of (V/U ) ; otherwise, (v, u) = T u (v) = T u ( v) = 0 for all v V, a contradiction. Here, v denotes the image of v in V/U. Notice that the bilinearity of the form means that T au1+bu 2 = at u1 + bt u2 where a, b C. This and the fact that T u is nonzero whenever u is nonzero implies that the map U (V/U ) given by u T u is an injective linear map. This gives the inequality m n dim(u ), or dim(u ) n m. On the other hand, choose a basis u 1, u 2,..., u m of U. Then U is the intersection of the kernels of the T ui. Each kernel has dimension n 1 and so the intersection of m such subspaces must have dimension at least n m. The two inequalities mean that dim U must be n m. Exercise 1.2. Fix a non-degenerate symmetric, bilinear form on V. Define the orthogonal group O(V ) to be the set of linear transformations g : V V that preserve the form, i.e. O(V ) := {g (v, w) = (g.v, g.w) for all v, w V. 1. Show that O(V ) is a subgroup of GL(V ), the group of invertible linear endomorphisms of V. 2. Suppose that we have another non-degenerate symmetric, bilinear form (, ) on V and define O (V ) using this form. Show that O(V ) and O (V ) are conjugate subgroups of GL(V ). In particular, they are isomorphic. (Hint: use the classification of such forms over C).

7 1.2. SOME KEY FACTS ABOUT LIE ALGEBRAS Jordan decomposition Recall that a matrix A is nilpotent if A k = 0 for some positive integer k. A matrix A is called semisimple if A has a basis of eigenvectors, i.e. A is diagonalizable 1. Theorem (Jordan decomposition for matrices). The Jordan decomposition says that every matrix A can be written uniquely as A = N + S where N is nilpotent, S is semisimple, and N and S commute. Moreover, there is another important property that is often used in Lie theory: a subspace U satsifies A(U) U if and only if S(U) U and N(U) U. The Jordan decomposition can be proved directly or it can be deduced from the Jordan canonical form of a matrix. The latter says that A is similar to a block diagonal matrix built out of Jordan blocks µ µ µ , µ µ where µ is an eigenvalue of A. Moreover the Jordan blocks that appear determine and are uniquely determined by the similarity class of A. In other words, A and B are conjugate under GL(V ) if and only if they have the same Jordan blocks in their Jordan decomposition. Exercise 1.3. In this exercise, we examine the nilpotent matrices, up to conjugation. 1. Show that a nilpotent matrix has only 0 as an eigenvalue and then use the Jordan canonical form to show that each conjugacy classes of nilpotent n n matrices corresponds to a unique partition of n. 2. Suppose that N is a nilpotent n n matrix with Jordan blocks of size λ 1 λ k. Compute the rank of N k in terms of the partition [λ i ]. 1.2 Some key facts about Lie algebras We will use the notation g for a Lie algebra. Recall that a Lie algebra is a vector space, which is equipped with a product [, ], called the bracket. The bracket is bilinear and antisymmetric. It also satisifies the Jacobi identity: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] for all X, Y, Z g. The prototypical Lie algebra is the endomorphisms of a vector space V, equipped with the bracket [X, Y ] = XY Y X, 1 Over a non-algebraically closed field, we would say that A is semisimple if it is diagonalizable over the algebraic closure of the field

9 1.2. SOME KEY FACTS ABOUT LIE ALGEBRAS Jordan decomposition The Jordan decomposition carries over to semisimple Lie algebras. An element S g is called semisimple if ad S is semisimple (as a linear map from the vector space underlying g to itself). Similarly, N g is nilpotent if ad N is nilpotent. Theorem (Jordan decomposition for g). For X g we can write X uniquely as X = X s + X n where X s g is semisimple, X n g is nilpotent and [X s, X n ] = 0. The element X s is called the semisimple part of X and X n is called the nilpotent part of X. The decomposition has two important properties: It behaves well under Lie algebra homomorphisms: if φ : g g is a Lie algebra homorphism, then φ(x) s = φ(x s ) and φ(x) n = φ(x n ). It coincides with the usual Jordan decomposition in gl(v ). In other words, whether we think of X gl(v ) as an element of the Lie algebra gl(v ) and use Theorem or we think of it as a regular old matrix and use Theorem 1.1.2, we get the same decomposition.

10 10 CHAPTER 1. BACKGROUND

11 Chapter 2 Jacobson-Morozov Theorem 2.1 sl 2 (C) The semisimple Lie algebra of smallest dimension is sl 2 (C), the 2 2 matrices of trace zero. It is of dimension three, with a basis given by [ ] [ ] [ ] E =, H =, F = These elements satisfy the relations [H, E] = 2E, [H, F ] = 2F, [E, F ] = H. (2.1) Notice that E and F are nilpotent matrices and so also nilpotent elements of sl 2 (C). And H is a semisimple matrix and a semisimple element of sl 2 (C). Or we could have seen this directly by writing the matrix for ad H in terms of the (ordered) basis E, H, F of g: ad H = Exercise 2.1. Write the matrices of ad E, ad F in terms of the basis E, F, H. Write the Cartan-Killing form using this basis and show that it is non-degenerate. This shows that sl 2 (C) is semisimple. On the other hand, it is easy to show that sl 2 (C) is simple directly. Here s the sketch, let I be an ideal. Then ad H (I) = I and so ad H restricted to I is semisimple (a fact from linear algebra). So I must be a sum of eigenspaces for H. Now use the action of E and F to show that if I is proper, then it must be zero. 2.2 Representations of sl 2 (C) The Lie algebra sl 2 (C) and its representations are the central building blocks in Lie theory. Recall that a representation of g is a Lie algebra homomorphism from g to gl(v ) for some vector space V. This amounts to finding matrices e, h, f gl(v ) satsifying the relations in 2.1. The classification of representations of sl 2 (C) has two components: 11

12 12 CHAPTER 2. JACOBSON-MOROZOV THEOREM Every representation is the direct sum of irreducible representations (this is true for all semisimple Lie algebras over the complex numbers). There is a unique irreducible representation (up to isomorphism) of dimension n for each positive integer n. 2.3 The irreducible representation of dimension n Let s now explicitly construct the irreducible representations of sl 2 (C). Consider the n n nilpotent matrix e = Next we seek a semisimple h such that [h, e] = 2. If we try to find h that is diagonal, we see that k k h = 0 0 k k 2n k 2n + 2 Exercise 2.2. Verify the above calculation for h. Show that if [e, f] = h, then h must have trace zero. Show that k = n 1 makes the trace of h equal to zero. Next to find f with [h, f] = 2f, we have that f must take the form a f = 0 a a n a n 1 0 Exercise 2.3. Show that [e, f] = h is satisfied exactly when a i = i(n i). Thus we have constructed a representation of sl 2 of dimension n. Verify that this is indeed an irreducible representation of sl 2 (C). Notice the following corollary of our construction: Corollary For every nilpotent matrix e gl n (C), there exists h, f gl n (C) satisfying the relations 2.1. This shows that every nilpotent matrix can be embedded in a copy of sl 2 (C) sitting inside of gl n (C) Proof. If we can do this for a nilpotent matrix e, then we can also do this for any conjugate geg 1 where g GL n (C). This is because we can conjugate h and f by g and obtain the necessary

13 2.4. JACOBSON-MOROZOV THEOREM 13 matrices. Hence we might as well take e to be in Jordan form. But for each Jordan block, we figured out the appropriate h and f above. If we do this for each block and put the blocks together, we get the relevant matrices h and f for e. For example, if (built from two Jordan blocks), then e = h = and work to satisfy the requirements of the corollary f = Jacobson-Morozov Theorem The corollary in the previous section holds in general for a semisimple Lie algebra g: Theorem (Jacobson-Morozov). Given a nilpotent element e g, there exists h, f g satisfying the relations 2.1. In other words, e belongs to a copy of sl 2 (C) sitting inside of g as a subalgebra. One proof of this use the result for gl n (C). Another proof to be sketched in the lectures is by induction on the dimension of g.

14 14 CHAPTER 2. JACOBSON-MOROZOV THEOREM

15 Chapter 3 Classifying nilpotent orbits The standing assumption is that g is semisimple. Our goal in these lectures is to classify the orbits of nilpotent elements in g under the action of the group G (of connected automorphisms of g). We can think of G either as an algebraic group (algebraic variety with a compatible group structure) or a Lie group (manifold with compatible group structure). 3.1 Dynkin-Kostant classification To classify the nilpotent G-orbits (that is, the nilpotent elements up to the action of G), we use the Jacobson-Morozov theorem: Let A hom denote the G-conjugacy classes of Lie algebra homomorphisms from sl 2 (C) to g. That is, two homomorphism φ, φ are conjugate if we there exists g G such that φ = phi Ad(g), where Ad(g) denotes the automorphism of g determined by g. Let H = [ ] and E = [ ] be elements of sl 2(C) as before. Consider the map Ω : A hom {nilpotent G-orbits in g} given by Ω(φ) = Ad(G)φ(E). This is a bijection: surjectivity is just the Jacobson-Morozov theorem and injectivity follows from a theorem of Kostant. Let Υ : A hom {semi-simple G-orbits in g} be the map Υ(φ) = Ad(G)φ(H). A theorem of Mal cev shows that Υ is injective. Conclusion: nilpotent orbits in g are parametrized by the image of Υ. This set was completely determined by Dynkin. The classification of nilpotent orbits in terms of these semisimple elements is called the Dynkin-Kostant classification. Pick a basis of simple roots {α 1,..., α n } for the root system of g with respect to a Cartan subalgebra h. Assume that h h belongs to a conjugacy class in the image of Υ. By using the action of the Weyl group, we can assume that α i (h) 0 for all i. Attaching these n numbers to the Dynkin diagram of g is what we call the weighted Dynkin diagram of the corresponding nilpotent orbit O. A few facts: α i (h) are integers. This follows from the representation theory of sl 2 (C). Moreover, α i (h) {0, 1, 2, }. This is one way to prove there are a finite number of nilpotent orbits. 15

16 16 CHAPTER 3. CLASSIFYING NILPOTENT ORBITS Let g i denote the i-eigenspace for ad h. Then another consequence of the representation theory of sl 2 (C) is that: dim(o) = dim g (dim g 0 + dim g 1 ). The dimension of g i can be computed from the weighted Dynkin diagram of O. For example, in G 2 we find (using techniques discussed later) that there are 5 nilpotent orbits. Their weighted Dynkin diagrams are and the dimensions of these orbits are 0 => 0, 0 => 1, 1 => 0, 2 => 0, 2 => 2 0, 6, 8, 10, 12. It is a general fact that orbits in g always have even dimension. 3.2 Bala-Carter Theorem to be continued...

17 Chapter 4 Classical groups 4.1 Parametrizing orbits by partitions 4.2 Partial order on orbits 17

### L(C G (x) 0 ) c g (x). Proof. Recall C G (x) = {g G xgx 1 = g} and c g (x) = {X g Ad xx = X}. In general, it is obvious that

ALGEBRAIC GROUPS 61 5. Root systems and semisimple Lie algebras 5.1. Characteristic 0 theory. Assume in this subsection that chark = 0. Let me recall a couple of definitions made earlier: G is called reductive

### INTRODUCTION TO LIE ALGEBRAS. LECTURE 10.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 10. 10. Jordan decomposition: theme with variations 10.1. Recall that f End(V ) is semisimple if f is diagonalizable (over the algebraic closure of the base field).

### GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS ANA BALIBANU DISCUSSED WITH PROFESSOR VICTOR GINZBURG 1. Introduction The aim of this paper is to explore the geometry of a Lie algebra g through the action

### LIE ALGEBRAS: LECTURE 7 11 May 2010

LIE ALGEBRAS: LECTURE 7 11 May 2010 CRYSTAL HOYT 1. sl n (F) is simple Let F be an algebraically closed field with characteristic zero. Let V be a finite dimensional vector space. Recall, that f End(V

### Classification of semisimple Lie algebras

Chapter 6 Classification of semisimple Lie algebras When we studied sl 2 (C), we discovered that it is spanned by elements e, f and h fulfilling the relations: [e, h] = 2e, [ f, h] = 2 f and [e, f ] =

### Lecture 11 The Radical and Semisimple Lie Algebras

18.745 Introduction to Lie Algebras October 14, 2010 Lecture 11 The Radical and Semisimple Lie Algebras Prof. Victor Kac Scribe: Scott Kovach and Qinxuan Pan Exercise 11.1. Let g be a Lie algebra. Then

### LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

### A CHARACTERIZATION OF DYNKIN ELEMENTS

A CHARACTERIZATION OF DYNKIN ELEMENTS PAUL E. GUNNELLS AND ERIC SOMMERS ABSTRACT. We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element

### ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

### 5 Quiver Representations

5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )

### Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

### is an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent

Lecture 4. G-Modules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of G-modules, mostly for finite groups, and a recipe for finding every irreducible G-module of a

### INTRODUCTION TO LIE ALGEBRAS. LECTURE 7.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 7. 7. Killing form. Nilpotent Lie algebras 7.1. Killing form. 7.1.1. Let L be a Lie algebra over a field k and let ρ : L gl(v ) be a finite dimensional L-module. Define

### Since G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =

Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for

### The Cartan Decomposition of a Complex Semisimple Lie Algebra

The Cartan Decomposition of a Complex Semisimple Lie Algebra Shawn Baland University of Colorado, Boulder November 29, 2007 Definition Let k be a field. A k-algebra is a k-vector space A equipped with

### LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F)

LECTURE 4: REPRESENTATION THEORY OF SL 2 (F) AND sl 2 (F) IVAN LOSEV In this lecture we will discuss the representation theory of the algebraic group SL 2 (F) and of the Lie algebra sl 2 (F), where F is

### Cartan s Criteria. Math 649, Dan Barbasch. February 26

Cartan s Criteria Math 649, 2013 Dan Barbasch February 26 Cartan s Criteria REFERENCES: Humphreys, I.2 and I.3. Definition The Cartan-Killing form of a Lie algebra is the bilinear form B(x, y) := Tr(ad

### Part III Symmetries, Fields and Particles

Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often

### MAT 445/ INTRODUCTION TO REPRESENTATION THEORY

MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations

### Exercises on chapter 1

Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

### 1.4 Solvable Lie algebras

1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)

### LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups

### THE THEOREM OF THE HIGHEST WEIGHT

THE THEOREM OF THE HIGHEST WEIGHT ANKE D. POHL Abstract. Incomplete notes of the talk in the IRTG Student Seminar 07.06.06. This is a draft version and thought for internal use only. The Theorem of the

### MATRIX LIE GROUPS AND LIE GROUPS

MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either

### REPRESENTATION THEORY, LECTURE 0. BASICS

REPRESENTATION THEORY, LECTURE 0. BASICS IVAN LOSEV Introduction The aim of this lecture is to recall some standard basic things about the representation theory of finite dimensional algebras and finite

### Symmetries, Fields and Particles. Examples 1.

Symmetries, Fields and Particles. Examples 1. 1. O(n) consists of n n real matrices M satisfying M T M = I. Check that O(n) is a group. U(n) consists of n n complex matrices U satisfying U U = I. Check

### Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).

9. Calogero-Moser spaces 9.1. Hamiltonian reduction along an orbit. Let M be an affine algebraic variety and G a reductive algebraic group. Suppose M is Poisson and the action of G preserves the Poisson

### A 2 G 2 A 1 A 1. (3) A double edge pointing from α i to α j if α i, α j are not perpendicular and α i 2 = 2 α j 2

46 MATH 223A NOTES 2011 LIE ALGEBRAS 11. Classification of semisimple Lie algebras I will explain how the Cartan matrix and Dynkin diagrams describe root systems. Then I will go through the classification

### A Little Beyond: Linear Algebra

A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of non-zero polynomials in [x], no two

### 16.2. Definition. Let N be the set of all nilpotent elements in g. Define N

74 16. Lecture 16: Springer Representations 16.1. The flag manifold. Let G = SL n (C). It acts transitively on the set F of complete flags 0 F 1 F n 1 C n and the stabilizer of the standard flag is the

### LIE ALGEBRAS: LECTURE 3 6 April 2010

LIE ALGEBRAS: LECTURE 3 6 April 2010 CRYSTAL HOYT 1. Simple 3-dimensional Lie algebras Suppose L is a simple 3-dimensional Lie algebra over k, where k is algebraically closed. Then [L, L] = L, since otherwise

### The Geometry of Conjugacy Classes of Nilpotent Matrices

The Geometry of Conjugacy Classes of Nilpotent Matrices A 5 A 3 A 1 A1 Alessandra Pantano A 2 A 2 a 2 a 2 Oliver Club Talk, Cornell April 14, 2005 a 1 a 1 a 3 a 5 References: H. Kraft and C. Procesi, Minimal

### Max-Planck-Institut für Mathematik Bonn

Max-Planck-Institut für Mathematik Bonn Cyclic elements in semisimple Lie algebras by A. G. Elashvili V. G. Kac E. B. Vinberg Max-Planck-Institut für Mathematik Preprint Series 202 (26) Cyclic elements

### ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

### ALGEBRA QUALIFYING EXAM PROBLEMS

ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General

### Problems in Linear Algebra and Representation Theory

Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific

### UC Berkeley Summer Undergraduate Research Program 2015 July 9 Lecture

UC Berkeley Summer Undergraduate Research Program 205 July 9 Lecture We will introduce the basic structure and representation theory of the symplectic group Sp(V ). Basics Fix a nondegenerate, alternating

### Representation Theory and Orbital Varieties. Thomas Pietraho Bowdoin College

Representation Theory and Orbital Varieties Thomas Pietraho Bowdoin College 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible

### Math 594. Solutions 5

Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

### MATH SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER Given vector spaces V and W, V W is the vector space given by

MATH 110 - SOLUTIONS TO PRACTICE MIDTERM LECTURE 1, SUMMER 2009 GSI: SANTIAGO CAÑEZ 1. Given vector spaces V and W, V W is the vector space given by V W = {(v, w) v V and w W }, with addition and scalar

### INTRODUCTION TO LIE ALGEBRAS. LECTURE 2.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined

### QUIVERS AND LATTICES.

QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given

### NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction

NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques

### 0 A. ... A j GL nj (F q ), 1 j r

CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected

### Mic ael Flohr Representation theory of semi-simple Lie algebras: Example su(3) 6. and 20. June 2003

Handout V for the course GROUP THEORY IN PHYSICS Mic ael Flohr Representation theory of semi-simple Lie algebras: Example su(3) 6. and 20. June 2003 GENERALIZING THE HIGHEST WEIGHT PROCEDURE FROM su(2)

### Classification of root systems

Classification of root systems September 8, 2017 1 Introduction These notes are an approximate outline of some of the material to be covered on Thursday, April 9; Tuesday, April 14; and Thursday, April

### arxiv: v1 [math.rt] 14 Nov 2007

arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof

### Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

### Topics in Representation Theory: Roots and Weights

Topics in Representation Theory: Roots and Weights 1 The Representation Ring Last time we defined the maximal torus T and Weyl group W (G, T ) for a compact, connected Lie group G and explained that our

### WOMP 2001: LINEAR ALGEBRA. 1. Vector spaces

WOMP 2001: LINEAR ALGEBRA DAN GROSSMAN Reference Roman, S Advanced Linear Algebra, GTM #135 (Not very good) Let k be a field, eg, R, Q, C, F q, K(t), 1 Vector spaces Definition A vector space over k is

### Exercises Lie groups

Exercises Lie groups E.P. van den Ban Spring 2009 Exercise 1. Let G be a group, equipped with the structure of a C -manifold. Let µ : G G G, (x, y) xy be the multiplication map. We assume that µ is smooth,

### REPRESENTATION THEORY. WEEKS 10 11

REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.

### Notation. For any Lie group G, we set G 0 to be the connected component of the identity.

Notation. For any Lie group G, we set G 0 to be the connected component of the identity. Problem 1 Prove that GL(n, R) is homotopic to O(n, R). (Hint: Gram-Schmidt Orthogonalization.) Here is a sequence

Algebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.

### COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA

COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties

### Linear and Bilinear Algebra (2WF04) Jan Draisma

Linear and Bilinear Algebra (2WF04) Jan Draisma CHAPTER 10 Symmetric bilinear forms We already know what a bilinear form β on a K-vector space V is: it is a function β : V V K that satisfies β(u + v,

### Simple Lie algebras. Classification and representations. Roots and weights

Chapter 3 Simple Lie algebras. Classification and representations. Roots and weights 3.1 Cartan subalgebra. Roots. Canonical form of the algebra We consider a semi-simple (i.e. with no abelian ideal) Lie

### Kac-Moody Algebras. Ana Ros Camacho June 28, 2010

Kac-Moody Algebras Ana Ros Camacho June 28, 2010 Abstract Talk for the seminar on Cohomology of Lie algebras, under the supervision of J-Prof. Christoph Wockel Contents 1 Motivation 1 2 Prerequisites 1

### Multiplicity free actions of simple algebraic groups

Multiplicity free actions of simple algebraic groups D. Testerman (with M. Liebeck and G. Seitz) EPF Lausanne Edinburgh, April 2016 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity

### Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

### Math 210C. A non-closed commutator subgroup

Math 210C. A non-closed commutator subgroup 1. Introduction In Exercise 3(i) of HW7 we saw that every element of SU(2) is a commutator (i.e., has the form xyx 1 y 1 for x, y SU(2)), so the same holds for

### SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS

1 SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS HUAJUN HUANG AND HONGYU HE Abstract. Let G be the group preserving a nondegenerate sesquilinear form B on a vector space V, and H a symmetric subgroup

### Representations of semisimple Lie algebras

Chapter 14 Representations of semisimple Lie algebras In this chapter we study a special type of representations of semisimple Lie algberas: the so called highest weight representations. In particular

### THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL LIE ALGEBRAS

Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 67 (2006) 2006, Pages 225 259 S 0077-1554(06)00156-7 Article electronically published on December 27, 2006 THE SEMISIMPLE SUBALGEBRAS OF EXCEPTIONAL

### A NOTE ON THE JORDAN CANONICAL FORM

A NOTE ON THE JORDAN CANONICAL FORM H. Azad Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia hassanaz@kfupm.edu.sa Abstract A proof of the Jordan

### Lie Algebras. Shlomo Sternberg

Lie Algebras Shlomo Sternberg March 8, 2004 2 Chapter 5 Conjugacy of Cartan subalgebras It is a standard theorem in linear algebra that any unitary matrix can be diagonalized (by conjugation by unitary

### Groups of Prime Power Order with Derived Subgroup of Prime Order

Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*

### PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS

PART I: GEOMETRY OF SEMISIMPLE LIE ALGEBRAS Contents 1. Regular elements in semisimple Lie algebras 1 2. The flag variety and the Bruhat decomposition 3 3. The Grothendieck-Springer resolution 6 4. The

### Representations of Matrix Lie Algebras

Representations of Matrix Lie Algebras Alex Turzillo REU Apprentice Program, University of Chicago aturzillo@uchicago.edu August 00 Abstract Building upon the concepts of the matrix Lie group and the matrix

### 2.4 Root space decomposition

34 CHAPTER 2. SEMISIMPLE LIE ALGEBRAS 2.4 Root space decomposition Let L denote a semisimple Lie algebra in this section. We will study the detailed structure of L through its adjoint representation. 2.4.1

### SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS

SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the

### Problem set 2. Math 212b February 8, 2001 due Feb. 27

Problem set 2 Math 212b February 8, 2001 due Feb. 27 Contents 1 The L 2 Euler operator 1 2 Symplectic vector spaces. 2 2.1 Special kinds of subspaces....................... 3 2.2 Normal forms..............................

### ALGEBRA 8: Linear algebra: characteristic polynomial

ALGEBRA 8: Linear algebra: characteristic polynomial Characteristic polynomial Definition 8.1. Consider a linear operator A End V over a vector space V. Consider a vector v V such that A(v) = λv. This

### (Kac Moody) Chevalley groups and Lie algebras with built in structure constants Lecture 1. Lisa Carbone Rutgers University

(Kac Moody) Chevalley groups and Lie algebras with built in structure constants Lecture 1 Lisa Carbone Rutgers University Slides will be posted at: http://sites.math.rutgers.edu/ carbonel/ Video will be

### Decay to zero of matrix coefficients at Adjoint infinity by Scot Adams

Decay to zero of matrix coefficients at Adjoint infinity by Scot Adams I. Introduction The main theorems below are Theorem 9 and Theorem 11. As far as I know, Theorem 9 represents a slight improvement

### Notes on Lie Algebras

NEW MEXICO TECH (October 23, 2010) DRAFT Notes on Lie Algebras Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA E-mail: iavramid@nmt.edu 1

### Math 5220 Representation Theory. Spring 2010 R. Kinser Notes by Ben Salisbury

Math 5220 Representation Theory Spring 2010 R. Kinser Notes by Ben Salisbury Contents Introduction Homework Assignments vii ix Chapter 1. Lie Algebras 1 1. Definitions 1 2. Classical Lie algebras 2 3.

### REPRESENTATION THEORY WEEK 5. B : V V k

REPRESENTATION THEORY WEEK 5 1. Invariant forms Recall that a bilinear form on a vector space V is a map satisfying B : V V k B (cv, dw) = cdb (v, w), B (v 1 + v, w) = B (v 1, w)+b (v, w), B (v, w 1 +

### Chapter 2 Linear Transformations

Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more

### Topics in linear algebra

Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over

### ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need

### ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS

ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,

### e j = Ad(f i ) 1 2a ij/a ii

A characterization of generalized Kac-Moody algebras. J. Algebra 174, 1073-1079 (1995). Richard E. Borcherds, D.P.M.M.S., 16 Mill Lane, Cambridge CB2 1SB, England. Generalized Kac-Moody algebras can be

### ABSTRACT ALGEBRA WITH APPLICATIONS

ABSTRACT ALGEBRA WITH APPLICATIONS IN TWO VOLUMES VOLUME I VECTOR SPACES AND GROUPS KARLHEINZ SPINDLER Darmstadt, Germany Marcel Dekker, Inc. New York Basel Hong Kong Contents f Volume I Preface v VECTOR

### 12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End

### Lemma 1.3. The element [X, X] is nonzero.

Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group

### Subgroups of Linear Algebraic Groups

Subgroups of Linear Algebraic Groups Subgroups of Linear Algebraic Groups Contents Introduction 1 Acknowledgements 4 1. Basic definitions and examples 5 1.1. Introduction to Linear Algebraic Groups 5 1.2.

### LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 18. Category O for rational Cherednik algebra We begin to study the representation theory of Rational Chrednik algebras. Perhaps, the class of representations

### (VII.B) Bilinear Forms

(VII.B) Bilinear Forms There are two standard generalizations of the dot product on R n to arbitrary vector spaces. The idea is to have a product that takes two vectors to a scalar. Bilinear forms are

### August 2015 Qualifying Examination Solutions

August 2015 Qualifying Examination Solutions If you have any difficulty with the wording of the following problems please contact the supervisor immediately. All persons responsible for these problems,

### Math 250: Higher Algebra Representations of finite groups

Math 250: Higher Algebra Representations of finite groups 1 Basic definitions Representations. A representation of a group G over a field k is a k-vector space V together with an action of G on V by linear

### Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification

### Topics in Representation Theory: SU(n), Weyl Chambers and the Diagram of a Group

Topics in Representation Theory: SU(n), Weyl hambers and the Diagram of a Group 1 Another Example: G = SU(n) Last time we began analyzing how the maximal torus T of G acts on the adjoint representation,

### The Morozov-Jacobson Theorem on 3-dimensional Simple Lie Subalgebras

The Morozov-Jacobson Theorem on 3-dimensional Simple Lie Subalgebras Klaus Pommerening July 1979 english version April 2012 The Morozov-Jacobson theorem says that every nilpotent element of a semisimple

### A Field Extension as a Vector Space

Chapter 8 A Field Extension as a Vector Space In this chapter, we take a closer look at a finite extension from the point of view that is a vector space over. It is clear, for instance, that any is a linear

### REPRESENTATIONS OF S n AND GL(n, C)

REPRESENTATIONS OF S n AND GL(n, C) SEAN MCAFEE 1 outline For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G Although

### Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

### Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition