CSA
MTH432 Central Simple Algebra
(August-November 2019)
Instructor : Dr Anupam Singh
Audience : 4th year BS-MS, iPhD, PhD students at IISER Pune
Schedule : Monday 10AM, Thursday 9AM, Friday 11AM, LHC 306
Evaluation : Test I - 20%, mid-sem - 30%, Test II - 20%, end-sem - 30%
Prerequisite : Basic Algebra courses (Groups, Vector spaves, Modules etc)
Goal of the course : This course is aimed at giving an introduction to some non-commutative objects which are very important in research.
Proposed Content : Semisimple and Simple modules, Semisimple rings, simple rings, Artin-Wedderburn theorem, Quaternion algebra, Central simple algebra, Cross-product construction, H^2, Brauer group, Application to Group algebra and character theory of finite groups.
Text books and reference material :
TIFR lecture notes on semisimple rings pdf file(thanks to Shripad Garge)
Farb & Dennis: Non-commutative algebra
Berhuy & Oggier: An introduction to central simple algebras and their applications to wireless communication
Pierce: Associative algebras
Lam: A first course in non-commutative rings
Mini Projects : This is a substitute for Test II.
Finite dimensional division algebras over finite fields and reals
Wedderburn's theorem on factorisation of polynomials
Br(Q) is infinte
Brauer's theorem for group algebra
Weekly topics covered :
1 August 19 - Rings (non-commutative with 1), (Left)-modules examples
2 August 19 - Algebra (associative) over R, examples
5 August 19 - Simple modules, Schur's Lemma
8 August 19 - Semisimple modules
9 August 19 - Tutorial Assignment-I
12 August 19 - Holiday (extra class) Semisimple ring, Artin-Wedderburn Theorem
15 August 19 - Holiday (extra class) Structure of simple Artinian rings
16 August 19 - Tutorial Assignment-II
19-30 August 19 --- no class
2 September 19 - Holiday
5 September 19 - Representation theory of finite groups using Artin-Wedderburn theorem
6 September 19 - Maschke's theorem, representations over arbitrary field.
9 September 19 - Tensor product of k-algebras, extension of scalars
12 September 19 - 2-sided ideals and center of tensor product of csa
13 September 19 -Tutorial Assignment-III
13 September 19 Test-I at 4pm
16 September 19 - Quaternion algebra over a field k
19 September 19 -When quaternion algebra is division?
20 September 19 - Tutorial
23 - 28 September 2019 Mid sem exam
Mid-sem exam on 25/9/2019 at 10AM.
30 September 2019 - Brauer equivalence on the set of finite dimensional central simple algebras over k vs. f.d. division algebras
03 October 2019 - Brauer group of a field k
04 October 2019 - Skolem-Noether Theorem
07 October 2019 - Centraliser theorem
10 October 2019 - Tutorial Assignment-IV
11 October 2019 - Dean's office substitutes it for Tuesday class
14-19 October : I am away for a conference
21 October 2019 - Holiday on the account of Election
24 October 2019 - Abelian Group Cohomology
25 October 2019 - Definitions H^1(G,A), H^2(G,A)
26 October 2019 - Extra class - presentations of projects
28 October 2019 - Maximal subfield, Splitting Field, Relative Brauer group
31 October 2019 - Br(K/k) --> H^2(Gal(K/k), K^*)
01 November 2019 - H^2(Gal(K/k), K^*) ---> Br(K/k)
04 November 2019 - Tutorial
07 November 2019 - Existence of Galois splitting field, Br(K/k) <--> H^2(Gal(K/k), K^*)
08 November 2019 - Group extension and H^2(G,A)
11 November 2019 - Hilbert 90, H^1 and forms
14 November 2019 - Cyclic algebras
15 November 2019 - Reduced norm, reduced trace, Decomposition
18 November 2019 - Tutorial
25-30 November End Sem exam
30 November 2019 - End Sem exam