This topics in algebra course will be a basic introduction to the theory of Lie algebras. The main reference
is the textbook ``Introduction to Lie Algebas and Representation Theory" by James E. Humphreys. It has not been ordered
for the UCSD bookstore, but copies may be purchased easily online. Another good text is ``Lie Algebras" by Nathan Jacobson.
We will cover some parts of chapters I-VI of the text, but certainly not all of this material. We will concentrate on Chapters I-III and hopefully the first parts of Chapters V and VI. The main goal is to describe the classification theory of semisimple Lie algebras. We will not necessarily follow the order of the material in Humphreys so closely.
The formal prerequisite for the course is Math 200, or an equivalent background in abstract algebra. Please see me if you are interested in taking the course but do not have this prerequisite. The grade in the course will be based primarily on attendance and homework. You may hand in any selection of the homework problems you wish to get feedback on, or you may discuss your ideas about the homework problems with me in office hours.
There will be no exams in this course. Please do come talk to me if you are registered for the course but feel that you will not be able to continue to attend regularly for whatever reason.
Below I will post brief summaries of what is covered in each lecture. After selected lecture summaries I will also list a few exercises from the text which I think would be good to go through to help cement your understanding of the material. It is important also to read the text and go over your lecture notes.
Lecture Summaries
Lecture 1 (1/06/14): Definition of a (not necessarily associative) algebra over a field F. Definition of a Lie algebra. Main example: given an
associative algebra A, the algebra with bracket [ab] = ab - ba is a Lie algebra. Applied to the matrix algebra M_n(F) this gives the general linear algebra
gl(n, F). The set of matrices of trace 0 is a subalgebra sl(n, F) of gl(n, F), the special linear algebra. Basic terminology for algebras is the same as for rings:
homomorphisms, ideals, factor algebras, etc. Definition of Abelian Lie algebra and the center of a Lie algebra. The 2-dimensional solvable Lie algebra. 5 minute outline of how Lie algebras are related to Lie groups (more details in a future lecture). Suggested exercises: 1.1, 1.2, 2.2, 2.3
Lecture 2 (1/08/14): Review of bilinear forms. Given any bilinear form ( , ) on a finite dimensional F-space V, the set L of matrices A
such that A^* = - A (where the adjoint A^* is the unique matrix such that (Av, w) = (v, A^* w) for all v, w) is a Lie subalgebra of gl(V). Choosing
a particular skew-symmetric form, L gives the symplectic algebra sl(V) (type C), whereas choosing a particular symmetric form gives the orthogonal algebra
(type B or D depending on whether V is odd or even dimensional). Type A is the special linear algebra. sl(2, F) is simple (in characteristic 0). Derivations.
Der A is a Lie subalgebra of gl(A) for any algebra A. The structure of Der F[x]. Suggested exercises: 1.8, 1.9, 1.10, 1.11
Lecture 3 (1/10/14): ad x is a derivation of a Lie algebra L. The set of inner derivations (those of the form ad x) is an ideal of Der L. ad: L to gl(L)
is a homomorphism of Lie algebras (the adjoint representation). The example of sl(2, F). Beginning definitions of solvable and nilpotent Lie algebras. The derived series
and lower central series. The examples of strictly upper triangular martrices (nilpotent) and upper triangular matrices (solvable). The 2-dimensional non-Abelian Lie algebra
is solvable, not nilpotent. Basic facts about solvable algebras L.
Definition of the radical rad L: the unique largest solvable ideal of L. Suggested exercises: 1.4, 1.6, 1.12, 2.4
Lecture 4 (1/13/14): Basic facts about nilpotent algebras L. Ad-nilpotent elements and algebras. If x is a nilpotent element of gl(V) then ad x is nilpotent. Main lemma along the way to Engel's theorem: A Lie subalgebra L of gl(V) consisting of nilpotent matrices has a common eigenvector
v in V with Lv = 0. Corrollary: any such L stabilizes a flag and thus consists of strictly upper triangular matrices with respect to some basis of V.
Suggested exercises: 3.3, 3.6, 3.7.
Lecture 5 (1/15/14): Engel's theorem: an ad-nilpotent algebra is nilpotent. If I is an ideal of a nilpotent Lie algebra L then I intersects Z(L) nontrivially.
From now on base field F characteristic 0 and algebraically closed. Lie's theorem: a solvable subalgebra of gl(V) is upper triangular with respect to some basis of V. Main lemma on the way to Lie's theorem:
a solvable subalgebra of gl(V) has a common eigenvector v in V. Suggested exercises: 4.1, 4.3.
Lecture 6 (1/17/14): Corollary of Lie's theorem: if L is solvable then its derived algebra [LL] is nilpotent. Cartan's criterion: if L is a subalgebra
of gl(V) and tr(xy) = 0 for all x in [LL], y in L, then L is solvable (assume without proof for now). Skip to Chapter 5: Semisimple algebras. The Killing form k(x,y) = tr(ad x ad y). Associative bilinear forms in general and Frobenius algebras. The Killing form is associative. A finite dimensional Lie algebra L is semisimple if and only if its Killing form is nondegenerate. Suggested exercises: 5.1, 5.2, 5.3, 5.6.
Lecture 7 (1/22/14): Every semisimple Lie algebra is a direct sum of uniquely determined simple ideals. If L is semisimple, and every derivation of L
is inner (of the form ad x). Definition of an L-module; modules and representations are differing language for the same concepts. All basic definitions and theorems (such
as the isomorphism theorems) for modules carry through from modules over associative algebras. An irreducible module V is one such that 0 and V are its only
submodules. Weyl's complete reducibility theorem: Every finite dimensional module over a finite dimensional semisimple Lie algebra is a direct sum of irreducible modules (proof
postponed). Suggested exercises: 6.2, 6.3, 6.4
Lecture 8 (1/24/14): Classification of irreducible sl(2,F)-modules (Chapter 7 of text): for each m there is a unique up to isomorphism
sl(2, F)-module of dimension m +1, which is a sum of 1-dimensional eigenspaces for h corresponding to the eigenvalues -m, -m+2, ..., m-2, m, and with
specified formulas for the action of x and y. Thus by complete irreducibility, we understand all finite dimensional sl(2, F)-modules. Suggested exercises: 7.2, 7.3, 7.4, 7.7
Lecture 9 (1/2714): If a faithful rep of a semisimple Lie algebra phi: L to gl(V) is fixed, there is a corresponding
nondegenerate symmetric associative bilinear form b(x, y) = tr(phi(x) phi(y)). The Casimir element c_phi is sum_i phi(x_i) phi(y_i),
where x_i runs over a basis of L and y_i is the dual basis under that bilinear form. the element c_phi commutes with all of phi(L),
and thus is an L-module homomorphism of V. Example: the Casimir element of sl_2 for the standard 2-dimensional rep. Schur's lemma.
Weyl's complete irreducibility theorem, first special case: if V is a f.d L-module where L is semisimple, and W is a codimension 1 irreducible submodule of V, then W has a complement. Suggested exercises: 6.1, 6.6, 6.7
Lecture 10 (1/29/14): Completion of the proof of Weyl's complete irreducibility theorem. First, if L is semisimple then any codimension-1 submodule of a f. d. L-module V has a complement. Then the main theorem: Any proper submodule of a f.d. L-module V
has a complement, and thus every f.d. module is a direct sum of irreducibles. Jordan decomposition: Every endomorphism x of a f.d. vector space V is uniquely a sum of a semisimple and nilpotent endomorphism which commute, and which are expressible as polynomials in x without constant term. Suggested exercises: 4.5, 4.6. Also: Prove that any set of semisimple endomorphisms of a f.d. vector space, all of which commute which each other, can be simultaneously diagonalized.
Lecture 11 (1/31/14): For a f.d. algebra A, the Lie algebra of derviations Der(A) is closed under taking semisimple and nilpotent parts. Since Der(L) = ad(L) for a semisimple Lie algebra L, using this one can introduce the abstract jordan decomposition x = s + n for x in L, such that ad x = ad s + ad n is the Jordan decomposition of ad x. Jordan decomposition is compatible with any representation phi: L to gl(V) in the sense that if x = s + n is the Jordan decomposition of x, then phi(x) = phi(s) + phi(n) is the Jordan decomposition in gl(V) (skip proof).
Beginning of the analysis of semisimple Lie algebras. Toral subalgebras. If H is a toral subalgebra of a semisimple L, it is abelian, and L has a corresponding root space decomposition as a direct sum of L_{alpha} where the alpha in H^* are roots. Here x in L_{alpha} if
[hx] = alpha(h) x for all h in H. First properties of this decomposition. Suggested exercises: 8.1, 8.3, 8.4
Lecture 12 (2/3/14): More on the root space decomposition: [L_{alpha}, L_{beta}] \subseteq L_{alpha+beta}. L_{alpha}
is orthogonal under the Killing form to L_{beta} unless alpha + beta = 0. H = C_L(H). The restriction of the Killing form to H is nondegenerate. The set of roots Phi spans H^*, and if alpha is a root, so is - alpha.
Lecture 13 (2/5/14): Finding copies of sl(2, F) inside a semisimple L. For each root alpha we can find x_{alpha} in L_{alpha},
y_{alpha} in L_{-alpha}, and h_{alpha} in H, such that these three elements span a subalgebra isomorphic to sl(2, F).
t_alpha is the element of H such that alpha = k(t_{alpha}, - ). For any x in L_{alpha}, y in L_{-alpha}, we have
[xy] = k(x,y) t_{alpha}. h_{alpha} = 2t_{alpha}/k(t_{alpha}, t_{alpha}). By studying how L as an S_{alpha}-module
decomposes as a direct sum of irreducibles and using our results on the structure of irreducibles over sl(2, F), one gets
that alpha, - alpha are the only roots that are scalar multiples of alpha. Suggested exercises: 8.5, 8.6, 8.7
Lecture 14 (2/7/14): By studying sum L_{beta + n alpha} as an S_{alpha}-module, one gets that
beta(h_{alpha}) \in Z and beta - beta(h_{alpha}) alpha is a root, for all roots alpha, beta. Also the roots of the form
beta + n alpha are beta - r alpha, ...., beta + q alpha, where q, r \geq 0 and r-q = beta(h_{alpha}). Fixing a basis of H^* given
by roots, alpha_1, ..., alpha_n, then the bilinear form ( , ) on H^* coming from the Killing form has rational coefficients, and any root beta has beta = sum c_i alpha_i where the c_i are rational. Then one can restrict the situation to the Q-vector space spanned
by the alpha_i, and finally by extending the base field we can work in the real vector space spanned by the alpha_i. then the set Phi
of roots is a root system in the Euclidean space sum R alpha_i with the positive definite bilinear form ( , ). Suggested Exercises: 8.8, 8.9, 8.11
Lecture 15 (2/10/14): Examples. For the Lie algebra Sp(4, F), full calculation of the root space decomposition and what the
root system is, following through the development in Chapter 8. Also for the Lie algebra sl(3, F). Suggested exercise: 8.2-- do for one
of the classical algebras whose maximal toral is 3-dimensional, such as the orthogonal algebra o(3, F).
Lecture 16 (2/12/14): Root systems. Possible angles between roots in a root system. Classification of root systems in the plane. The Weyl group.
Idea of a base of a root system. Lemma, if alpha and beta are nonproportional roots, then in (alpha, beta) > 0 then alpha -beta is a root, while if (alpha, beta) < 0
then alpha + beta is a root. Suggested exercises: 9.4, 9.5, 9.6, 9.7
Lecture 17 (2/14/14): Every root system has a base. More precisely, given a root system Phi, choose a vector gamma in E which
is regular in the sense that (gamma, alpha) \neq 0 for all alpha in Phi. Then there is a base Delta = Delta(gamma) where Delta is the set of
indecomposable elements in the set Phi^+(\gamma) of roots beta such that (gamma, beta) > 0 (and then Phi^+(gamma) is the set of positive roots
with respect to this base.) Moreover, every base has the form Delta(gamma) for some gamma. Lemma: if sigma_{\alpha} is a reflection
where alpha in Delta is in the base, then sigma_{\alpha} permutes the positive roots other than alpha. Suggested exercises: 10.2, 10.7.
Lecture 18 (2/19/14): Given any two bases of the root system, some element of the Weyl group sends one to the other. The Weyl group
is generated by simple reflections. The Cartan matrix of a root system. Dynkin diagrams. Setup for the classification of Dynkin diagrams.
Lecture 19 (2/21/14): Classification of Dynkin diagrams A-G (partial proof, some steps omitted). Isomorphism of root systems. Any two root systems having
the same Dynkin diagram (or same Cartan matrix) are isomorphic root systems. Suggested exercises: 11.4, 11.6.
Lecture 20 (2/24/14): Some parts of the theory we will not give the full proof of: The root system of a semisimple Lie algebra L doesn't depend on the choice of maximal toral H. Semisimple Lie algebras with isomorphic root systems are isomorphic as algebras. There does exist a simple Lie algebra with root system equal to each of the irreducible root systems we classified; in particular, the classical algebras give root systems of types A-D. Next, brief introduction to Lie groups and their relationship to Lie algebras. Definition of Lie group. Examples. A vector field is a derivation of the ring of C-infinity functions on a smooth manifold M. Brief idea of how this relates to the idea of choosing a vector in the tangent space at each point,
such that these vectors vary smoothly. The bracket of vector fields is a vector field.
Lecture 21 (2/26/14): If G is a Lie group, one can look at left invariant vector fields. There is exactly one of these for each choice of a tangent vector
in the tangent space to the identity element, so the left invariant vector fields form a finite-dimensional vector space. The bracket of left invariant vector fields
is left invariant, so this is a Lie subalgebra of the Lie algebra of all vector fields. This is the finite-dimensional Lie algebra g assigned to the Lie group G. Example:
if G = (R^d, +) then g is just the abelian lie algebra. One can use this correspondence to classify simple Lie groups, since their Lie algebras are also simple.
Brief introduction to universal enveloping algebras. The free associative algebra. If L is a f.d. Lie algebra over F,
with basis x_1, dots, x_n, then U(L) is k < x_1, dots x_n>/I where I is generated by relations x_j x_i - x_i x_j - [x_ix_j] for all i < j.
So there is a map L to U(L), which is a Lie-algebra homomorphism when U(L) is considered as a Lie algebra under the commutator. L \to U(L) is universal
for maps from L to associative algebras R which are Lie algebra homomorphisms when R is considered as a Lie algebra under the commutator. The PBW theorem (omit proof).
Suggested exercises: 17.1, 17.3.
Lecture 22 (2/28/14): The universal enveloping algebra U(L) has a filtration such that the associated graded ring is isomorphic
to a polynomial ring. L-modules are equivalent to U(L)-modules. Beginning of the study of modules V over a semisimple Lie algebra L, with maximal toral
H, root system Phi and base of positive roots Delta. A weight space for mu in H^* is V_{mu} = { v | hv = mu(h) v for all h in H}. The direct sum
of the weight spaces is a submodule of V, which is nonzero if V is f.d. So if V is f.d. and irreducible, it is the direct sum of its weight spaces.
Lecture 23 (3/3/14): Standard cyclic (or highest weight) modules: those L-modules generated by a weight vector v of weight lambda such that
v is killed by all x in L_{alpha}, alpha a positive root. The structure of such modules: (1) every element is obtainable from a product
of y_{\alpha}'s applied to v; (2) V is the direct sum of its weight spaces, and all weights mu occurring are less than or equal to lambda in the ordering, i.e.
each mu is equal to lambda minus a sum of positive roots; (3) all weight spaces of V are finite-dimensional; (4) Every submodule of V is the direct sum of its
weight spaces, and is contained in the sum of weight spaces other than V_{lambda} if it is proper; (5) V has a unique proper submodule and unique irreducible factor
module; (6) every factor module of V is also a standard cyclic module.
Suggested exercises: 20.1, 20.2, 20.3.
Lecture 24 (3/5/14): (7). The maximal vector in a standard cyclic module is unique up to scalar. Any two irreducible standard cyclic modules of
highest weight lambda are isomorphic. There does exist a (unique up to isomorphism) irreducible standard cyclic module of weight lambda, call this V(lambda).
Any f.d. irreducible module is one of the V(lambda), but not all V(lambda)'s are finite dimensional. If V(lambda) is finite dimensional, we get
< lambda, alpha> = 2(lambda, alpha)/(alpha, alpha) is a nonnegative integer for all positive roots alpha.
Lecture 25 (3/7/14): Discussion of the intergral weight lattice and dominant integral weights. If V(lambda) is f.d., then lambda
is dominant integral. Theorem: if lambda is dominant integral, then V = V(lambda) is f.d. Also, the set of weights for V is permuted by the Weyl group,
with a weight space V_{mu} having the same dimension as V_{sigma mu} for all sigma in the Weyl group.
Lecture 26 (3/10/14): Beginning of the proof of the theorem (21.2 in text)
Lecture 27 (3/12/14): Conclusion of the proof of 21.2. Some examples of the weights occurring for V(lambda) for several choices of
dominant integral lambda, where L = sl(3, F).
Lecture 28 (3/14/14): Cookies. Discussion of recent work of Sierra and Walton: "The universal enveloping algebra of
the Witt algebra is not noetherian".