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Quantum Groups Lie Groups and Quantum Groups Prof. Vera Serganova notes by Theo Johnson-Freyd UC-Berkeley Mathematics Department Spring Semester 2010 Contents Contents 1 Introduction 4 0.1 Conventions and numbering................................ 5 Lecture 1 Jan 20, 20105 1.1 ................................................ 5 1.2 Root and Weight lattices.................................. 7 Lecture 2 Jan 22, 20109 2.1 Weight Modules ...................................... 9 2.2 Weyl Character Formula.................................. 10 2.3 Verma module ....................................... 11 Lecture 3 Jan 25, 2010 12 3.1 Casimir element ...................................... 13 3.2 Category O (BGG category) ............................... 14 Lecture 4 Jan 27, 2010 16 4.1 Kostant partition function................................. 16 4.2 BGG resolution....................................... 17 4.3 Dimension formula..................................... 17 4.4 Application to tensor product............................... 18 Lecture 5 Jan 29, 2010 19 5.1 Center of universal enveloping algebra .......................... 20 5.2 Harish-Chandra homomorphism.............................. 22 1 Lecture 6 Feb 1, 2010 22 Lecture 7 Feb 3, 2010 25 7.1 Nilpotent cone ....................................... 27 Lecture 8 Feb 5, 2010 28 Lecture 9 Feb 8, 2010 31 9.1 G as an algebraic group .................................. 31 Lecture 10 Feb 10, 2010 34 10.1 General facts about algebraic groups........................... 35 10.2 Compact Groups...................................... 36 Lecture 11 Feb 12, 2010 38 11.1 Unitary Representations.................................. 38 11.2 Cartan decomposiiton ................................... 40 Lecture 12 Feb 5, 2010 40 12.1 Compact Groups...................................... 41 12.2 Real representation theory................................. 43 12.3 ................................................ 44 Lecture 13 Feb 19, 2010 44 13.1 Poisson Algebra....................................... 44 13.2 Coadjoint orbits for compact groups ........................... 45 13.3 G=P as a projective algebraic variety........................... 46 Lecture 14 Feb 22, 2010 47 14.1 Homogeneous spaces.................................... 47 14.2 Solvable groups....................................... 49 Lecture 15 Feb 24, 2010 51 15.1 Parabolic Lie algebras ................................... 52 Lecture 16 Feb 26, 2010 53 16.1 Flag manifolds for classical groups ............................ 54 16.2 Bruhat decomposition ................................... 55 Lecture 17 March 1, 2010 56 17.1 Bruhat order ........................................ 58 17.2 Geometric induction.................................... 58 Lecture 18 March 3, 2010 59 18.1 Frobenius reciprocity.................................... 59 2 Lecture 19 March 5, 2010 63 Lecture 20 March 8, 2010 66 Lecture 21 March 10, 2010 70 Lecture 22 March 12, 2010 73 Lecture 23 March 15, 2010 75 Lecture 24 March 17, 2010 78 24.1 Cohomology of Lie algebras................................ 79 Lecture 25 Feb 5, 2010 82 Lecture 26 March 29, 2010 85 26.1 Kostant theorem ...................................... 85 26.2 Quantum Groups...................................... 87 Lecture 27 March 31, 2010 89 27.1 SLq(2) and GLq(2)..................................... 90 Lecture 28 April 2, 2010 93 28.1 Classical limit (when q = 1)................................ 93 28.2 Finite-dimensional irreducible representations of Uqsl(2)................ 93 Lecture 29 April 5, 2010 95 Lecture 30 April 7, 2010 98 30.1 If q is a root of unity (degree l)..............................100 Lecture 31 April 9, 2010 101 31.1 Kac-Moody Lie algebras..................................101 31.2 Quantum enveloping algebras for finite-dimensional semisimple Lie algebras . 102 Lecture 32 April 12, 2010 104 Lecture 33 April 14, 2010 107 Lecture 34 April 16, 2010 110 Lecture 35 April 19, 2010 113 Lecture 36 May 21, 2010 116 36.1 R-matrices and the group algebra.............................118 3 Lecture 37 April 23, 2010 120 37.1 Quantum function algberas ................................121 Lecture 38 April 26, 2010 122 38.1 Kashiwara crystal bases ..................................122 Lecture 39 April 28, 2010 125 Lecture 40 April 30, 2010 128 References 133 Index 134 Introduction These are notes from UC Berkeley's Math 261B, taught by Vera Serganova in the Spring of 2010. The class met three times a week | Mondays, Wednesdays, and Fridays | from 1pm to 2pm. Needless to say, the content is due to VS, and all the errors are due to me, the notetaker (Theo Johnson-Freyd). VS's website for the course is at http://math.berkeley.edu/~serganov/261/ index.html. This course is a continuation of the Fall 2009 course Math 261A: Lie Groups, taught by Prof. Ian Agol. IA's website for that course is http://math.berkeley.edu/~ianagol/261A.F09/. I don't know of notes from that course. Edited notes from previous versions of 261A are [1,5]. Unedited notes from a previous version of 261B are [11]. As with my other course notes, I typed these mostly for my own benefit, although I do hope that they will be of use to other readers. (It was Anton's excellent notes from a variety of classes | in addition to the Lie groups notes mentioned above, he has other notes on his website | that inspired me to type my own notes, and I have borrowed from his preamble.) I apologize in advance for any errors or omissions. Places where I did not understand what was written or think that I in fact have an error will be marked **like this**. Please e-mail me (mailto: [email protected]) with corrections. For the foreseeable future, these notes are available at http://math.berkeley.edu/~theojf/QuantumGroups10.pdf. These notes are typeset using TEXShop Pro on a MacBook running OS 10.6; the backend is pdfLATEX. Pictures are drawn using pgf/TikZ. The raw LATEX sources are available at http: //math.berkeley.edu/~theojf/QuantumGroups10.tar.gz. These notes were last updated May 2, 2010. 4 0.1 Conventions and numbering Each lecture begins a new \section", and if a lecture breaks naturally into multiple topics, I try to reflect that with subsections. Equations, theorems, lemmas, etc., are numbered by their lecture. Theorems, lemmas, propositions, corollaries, and examples are counted with the same counter. Definitions are not marked qua definitions, but italics always mean that the word is being defined by that sentence. All definitions are indexed in the index. A list of all theorems, propositions, etc., is also at the end of the document. To generate these lists and to format theorems, etc., I have used the package ntheorem. Better referencing is done by cleveref. Lecture 1 Jan 20, 2010 VS's website is math.berkeley.edu/~serganov/. Office hours are Monday 4-5:30 and W 11-12:30, in 709 Evans. This course is on Lie groups and quantum groups. We will talk about: 1. Representation theory of semisimple Lie algebras: (a) Algebraic Methods: Weyl character formula, Harish-Chandra theorem, category O(BGG), cohomology, Zuckerman functor. (b) Geometric Methods: The ultimate goal is Beilinson-Bernstein. To move to this, we will study localization, flag varieties, nilpotent cone, Springer resolution, Borel-Weil-Bott theorem. And a little bit about the orbit method for nilpotent groups. 2. Quantum Groups. Hopf algebras, tensor categories, quantization. The main result is the existence of canonical bases. (We may or may not have time to talk about roots of unity.) Question from the audience: Is there a good reference? Answer: We will not follow a particular book, but VS will post references on the website. And Theo is taking notes. Some good references are [4, 10,2,3,7,8,6]. 1.1 We begin with something that you should know, but we will fix notation. By g we mean a semisimple (later, reductive) Lie algebra over C. By G we mean a simply-connected connected group with Lie algebra g. There are several ways to look at this group. If you think of it as a Lie group really, then its representation theory is very rich. But usually we mean it as an algebraic group. Maybe you covered this in 261A: there is a unique compact simply connected real Lie group K with complexification KC = G. So we will not spend much time on compact groups, because everything follows from the complex presentations. 5 Let's then recall the root decomposition and Cartan subalgebra. We pick a subalgebra h ⊆ g, which is maximal abelian consisting of semisimple elements. Then the action of h on g is semisimple, so we have the root decomposition M g = h ⊕ gα α2∆ ∗ where ∆ ⊆ h is the root system, and gα = fx 2 g s.t. [h; x] = α(h)xg. Then dim gα = 1. This follows from 261A. Example 1.1 Let g = sln+1 ⊆ gln+1. We take coordinates 0; : : : ; n given by i(aij) = aii, and h is a diagraonal subalgebra. Indeed, ∆ = fi − j s.t. i 6= jg, and gi−j = CEij. ♦ Let W be the Weyl group, which is a linear group acting on h∗, generated by all the root reflections sα. In the above example, W = Sn+1. In general, ∼ W = NG(h)=H Example 1.2 The group G2 is an exceptional 14-dimensional Lie algebra. Its root diagram is: · · • · · • • • • · • • • · • • • • · · • · · ♦ We recall the triangular decomposition, where we pick a hyperplane that does not pass through any roots, and so ∆ = ∆+ t ∆−, where ∆+ = fα 2 ∆ s.t. (α; γ) > 0g for some generically chosen γ. As soon as we have this decomposition, we can construct the triangular decomposition g = n− ⊕ − ± L ± + h ⊕ n , where n = α2∆± gα. Then n are nilpotent subalgebras, and b = h ⊕ n are solvable subalgebras, and we have: Theorem 1.3 Every maximal solvable subalgebra of g is conjugate to b. We can choose a basis α1; : : : ; αn 2 ∆ of simple roots such that every positive root is: X α = miαi; mi 2 Z≥0 + In sln+1, with the standard choice ∆ = fi − j s.t. i > jg, the simple roots are i − i+1 for i = 0; : : : ; n − 1. We define the rank of g to be dim h. Any α 2 ∆ gives rise to an sl2-subalgebra inside g. How? We pick xα 2 gα and yα 2 g−α, and hα 2 h such that [hα; xα] = 2xα,[hα; yα] = −2yα, and [xα; yα] = hα.
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