Lie Algebras and Representation Theory Course Description

ALGEBRAIC STRUCTURES: A SECOND YEAR ALGEBRA SEQUENCE

JULIA PEVTSOVA AND PAUL SMITH

1. 1st quarter: Lie algebras and Representation theory

Course description. This course will cover the foundations of semi-simple Lie algebras, and their representation theory. It will also lay foundations for any further study of representation theory, Lie theory, and many other related topics. We shall discuss the structure of concrete examples of classical Lie algebras such as gln, sln, spn and son. This will be followed by the general theory of universal enveloping algebras and PBW theorem. The abstract theory of root systems and weight lattices that we shall develop will allow us to classify semi-simple Lie algebras. The second part of the course will be devoted to representations of Lie algebras and the theory of weights. Most of the theory will be developed over the complex numbers. Additional topics to discuss, time permitting, will be real Lie algebras and Lie algebra coho- mology.

Topics at a glance. (1) Deﬁnitions and examples (2) Universal enveloping algebra and PBW theorem (3) Solvable and nilpotent Lie algebras (4) Root systems (5) Classiﬁcation of semi-simple Lie algebras (6) Serre relations (7) Representation theory: (a) Highest weight modules (b) Finite dimensional irreducible modules, Dominant weights (c) Weyl character formula (8) Chevalley basis Reference. J. Humphreys, “Introduction to Lie algebras and Representation theory”.

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2. 2nd quarter: Non-commutative Algebras

Course description. This course will cover the foundations of finite- and infinite- dimensional associative algebras. The basic finite-dimensional examples will be the group algebra of a finite group, and the path algebra of appropriate quivers with relations. The basic infinite-dimensional examples will be the enveloping algebras of Lie algebras, rings of differential operators. and related algebras. The finite dimensional theory will build on the material covered in the first-year algebra course: simple modules, Schur’s lemma, composition series, the Jacobson radical, semisimple rings and modules, the Artin-Wedderburn Theorem. Among the examples discussed in detail will be the representation theories of the group SL(2, Fp), the Lie algebra sl(2, Fp), the Kronecker quiver, and the quivers of types A, D, E.

Topics at a glance. (1) Finite Dimensional Algebras and Artinian Rings (a) Indecomposable modules (b) Local endomorphism rings (c) The Krull-Schmidt Theorem (d) The bijection between indecomposable projectives and simple modules. (e) The Cartan matrix. (f) Blocks (g) The quiver associated to a finite dimensional algebra (h) Path algebras of quivers (i) Representation type; finite/tame/wild trichotomy (j) Gabriel’s Theorem (2) Infinite Dimensional Algebras (a) Bergman’s Diamond Lemma (b) Gelfand-Kirillov Dimension (c) Localization in non-commutative rings — the Ore condition (d) Goldie’s Theorem (e) Maximal Orders (f) Rings finite over their centers. Artin-Tate Lemma. Examples: Enveloping algebras in positive characteristic. • (X) o G, G a finite group acting on affine variety X. • OSome path algebras of quivers with relations. • References. J.L. Alperin, “Local Representation Theory” J. Humphreys, “Introduction to Lie algebras and Representation theory” R.S. Pierce, “Associative Algebras” ALGEBRAIC STRUCTURES: A SECOND YEAR ALGEBRA SEQUENCE 3

3. 3rd quarter: Categorical and homological structures and methods

Course description. This course will be a must for anyone thinking about algebra, algebraic topology, algebraic geometry or representation theory as a possi- ble research direction. Topics include some category theory: homological algebra, abelian and triangulated categories with multiple examples (derived categories, sta- ble module categories, spectra), Monoidal categories, Morita equivalence and tilting theory, Grothendieck group. (1) Homological invariants of rings (2) Koszul algebras (3) Hochschild (co)homology. Meaning of low-dimensional groups. (4) The Grothendieck group of a ring/exact category. The trace map from K-theory to Hochschild homology. (5) Monoidal categories – Hopf algebras (6) Functors between module categories: Bimodules and Watt’s Theorem • Progenerators and Morita equivalence • Tilting modules (7) Triangulated• categories (8) Derived categories Reference. C. Weibel, “Introduction to homologial algebra”. 4 JULIAPEVTSOVAANDPAULSMITH

4. Comments to ourselves (1) I like the treatment of Goldie’s Theorem in P.M. Cohn’s ”Algebra, Vol. 2.” (PS) (2) There is quite a bit of material that I would like to see developed in the exercises during the second quarter. For example: The Jacobson Density Theorem: let M be a simple left R-module and • D its endomorphism ring. If m1,...,mn are linearly independent over 0 0 D and m1,...,mn are arbitrary elements of M, then there exists an 0 x R such that xmi = mi for all i. Bimodules∈ and Watt’s Theorem: if F : ModR ModS is a right exact • → functor that commutes with direct sums, then F ∼= B R where B is an S-R-bimodule ⊗ − A left artinian ring R is Morita equivalent to a ring S such that S/J(S) • is a product of division rings. The center of a ring is a Morita invariant: it is isomorphic to the • endomorphism ring of the identity functor on the module category. The dual basis lemma for projective modules: if p1,...,pn generate • a projective left R-module P , then there are elements f1,...,fn ∨ ∈ P := HomR(P,R) such that p = f1(p)p1 + +fn(p)pn for all p P . the natural map P P ∨∨ is an isomorphism··· for finitely generated∈ • projectives → A finitely generated algebra over a field k is artinian if and only if it is • finite dimensional. (Ex. Atiyah-Macdonald) I don’t see how to prove this (PS) Let I be an ideal in a left noetherian ring. Then I contains a product • of prime ideals; there are finitely many minimal primes containing I; if √I denotes the intersection of the minimal primes containing I, then (√I)n I for n 0. Artin-Tate⊂ Lemma: if R is a finitely generated k-algebra and Z is a • central subalgebra of R such that R is a finitely generated Z-module, then Z is a finitely generated k-algebra. Hence Z and R are noetherian. Indecomposable injectives, uniform modules, injective envelopes of • simple modules. The Coxeter transformation for a finite dimensional algebra of finite • global dimension, c : K0(A) K0(A), defined b y c([Pj]) = [Ij ] → − where Pj and Ij are the projective cover and injective hull of the simple module Sj . (3)