## Mathematics 534, section 101. Lie Theory I

### Term 1, 2013, TTh 11:00 - 12:30, Math Annex 1102

• Instructor :
 Zinovy Reichstein Office: 1105 Math Annex Phone: 2-3929 E-mail: reichst at math dot ubc dot ca

• Textbook:
 James E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972.
• Course description:
 Lie theory is the study of continuous group of tranformations. These groups play an important role in various areas of mathematics, from PDEs to number theory, as well as in physics. Their structure is most easily understood by in studying their ``linear approximations", otherwise known as Lie algebras. This course we will focus on the study of finite-dimenional Lie algebras and their representations by algebraic methods. We will discuss nilpotent, solvable, and semisimple Lie algebras, classify the root systems, talk about weights, highest weight modules, and universal enveloping algebras. Our ultimate goal will be the classification of simple complex Lie algebras. This material is foundational for many areas of pure mathematics. Our textbook is concise and beautifully written. I plan to follow it closely through much of the term, and cover most of the material in it. Next term's follow up class, Math 535 (Lie Theory II) will focus on the theory of algebraic groups. It will be taught by Julia Gordon.
• Prerequisites:
 High comfort level with linear algebra, including Jordan canonical form of a matrix. Familiarity with abstract algebra will also be helpful.
• Homework, class projects and evaluation:
 I plan to assign problem sets roughly every other weeek. Each student will be expected to complete a class project on a topic related to Lie Theory and write an expository paper on it during the term. Depending on the nature of the project and the size of the class, I will also ask some (and possibly all) students to present the highlights of their projects in class. The final course mark will be based on homework and this project.
• Problem Set 1, due Tuesday, Sept. 10.
 Page 5. Problems 1, 2, 3, 6, 10, 12. Problem A: Show that the Lie algebra sl_n is simple. In other words, in this algbera has no ideals, other than (0) and sl_n. Assume that the characteristic of the base field does not divide n.
• Problem Set 2, due Tuesday, October 1.
 Page 14. Problems 2, 5, 6, 7, 10 Page 20. Problems 1, 3, 4
• Problem Set 3, due Thursday, October 24.
 Page 24. Problems 2, 6 (Hint: show that the algebra in question is simple, then use problems 6, 7 on p. 31). Pages 30--31. Problems 2, 6, 7. In problems B and C below L is assumed to be a finite-dimensional Lie algebra (not necessarily semisimple). By a "module" we mean a finite-dimensional L-module. Recall that a module is called completely reducible if it is a direct sum of irreducible modules. The base field F is assumed to be algebraically closed of characteristic 0 throughout. Problem A: Prove formula (*) used in the proof of Lemma 4.2(b) on pp. 18-19. Problem B: Show that if a module V is completely reducible then so is every submodule and every factor-module of V. Problem C: Show that if V and W are completely reducible modules then so is the tersor product of V and W (over F).
• Problem Set 4, due Thursday, November 14.
 Pages 40-41. Problems 4, 5, 7, 10 Pages 45-47. Problems 3, 9, 11 Problem A: Show that any subset of E \ { 0} satisfying condition (R4) on p. 42 is automatically finite.
• Problem Set 5, due Thursday, November 28.
 Pages 5--6. Problem 10. Assume the base field is the field of complex numbers, F = C. Ignore the question about D_1. Hint: Exhibit a maximal toral subalgebra and find the associated root system. Use this data to show that the algebra is simple in each case. Then appeal to the Isomorphism Theorem 14.2. Page 63. Problems 3, 4. Pages 66--67. Problem 4.
• Project ideas:
 1. Ado-Iwasawa Theorem: Every finite-dimensional Lie algebra is isomorphic to a subalgebra of gl_n. I would be content with Ado's part (in char. 0), but if someone wants to go through Iwasawa's proof as well, that would be great. Possible source, N. Jacobson, Lie algebras, Chapter VI. N. Bourbaki, Lie groups and Lie Algebras, Chapter I. 2. Varieties of structure constants. Possible source: Yu.A. Neretin, An estimate for the number of parameters defining an n-dimensional algebra, Math. USSR-Izv. 30 (1988), no. 2, 283--294. 3. Asymptotic formulas for the number of groups of order p^r. Possible sources: G. Higman, Enumerating p-groups. I, Proc. London Math. Soc. (3) 10 (1960), 24--30. C. C. Sims, Enumerating p-groups. Proc. London Math. Soc. (3) 15 (1965), 151--166. S. Blackburn, P. Neumann, G. Venkataraman, Enumeration of finite groups. Cambridge Tracts in Mathematics, 173. Cambridge University Press, 2007. 4. Free Lie algebras, Campbell-Hausdorff formula. Possible source: J.-P. Serre, Lia algebras and Lie groups, Chapter IV. 5. Basic theory of quadratic forms. Possible source: T.Y. Lam, Introduction to quadratic forms over fields, Chapter I. 6. Define Lie algebra cohomology groups and prove Whitehead's theorem: if L is a semisimple complex Lie algebra and M is a noni-trivial irreducible module then H^i(L, M) = (0). Deduce Weyl's theorem and Whitehead's lemmas as corollaries. Possible sources: Jacobson, Lie algebras, Chapter 4. Weibel, An introduction to homological algebra, Chapter 7. See also Notes by Friedrich Wagemann 7. Definition of Kac-Moody Lie algebras. Possible source: Moody, Pianzola, Lie Algebras with Triangular Decompositions. 8. Prove Mal'tsev's theorem: Suppose L_1 and L_2 are two Levi subalgebras in a finite-dimensional algebra L. Then there exists an automorphism of L of the form f = exp(ad(x)) such that x is an element of the nilpotent radical of L and f takes L_1 to L_2. Possible sources: N. Bourbaki, Lie groups and Lie Algebras, Chapter I, A. L. Onishchik, E. B. Vinberg, Lie groups and algebraic groups. Springer-Verlag, Berlin, 1990, Chapter 6. 9. The existence of a maximal compact subgroup in a complex reductive group. (The connection is via Lie algebras.) Possible sources: A. L. Onishchik, E. B. Vinberg, Lie groups and algebraic groups. Springer-Verlag, Berlin, 1990, Chapter 5. A. Kirillov, Jr., An introduction to Lie groups and Lie algebras. Cambridge University Press, 2008. Section 6. 10. Kirillov's orbit method. This was originally used to describe representations of nilpotent Lie groups. It has since been extended to other classes of groups. The case where G is a finite p-group may be particularly accessible. Here are some references. MR0142001 (25 #5396) Kirillov, A. A. Unitary representations of nilpotent Lie groups. (Russian) Uspehi Mat. Nauk 17 1962 no. 4 (106), 57–110. (English translation should be available in Russian Math. Surveys.) MR1701415 Kirillov, A. A. Merits and demerits of the orbit method. Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 433–488. MR2166924 Kirillov, A. A. The orbit method and finite groups. Surveys in modern mathematics, 34–69, London Math. Soc. Lecture Note Ser., 321, Cambridge Univ. Press, Cambridge, 2005. MR2801175 (2012e:20014) Kamgarpour, Masoud; Thomas, Teruji Compatible intertwiners for representations of finite nilpotent groups. (English summary) Represent. Theory 15 (2011), 407–432.