Algebra-I

Aug-Dec 2018 (09:30-11:00 Mondays, Wednesdays, Fridays)

Books:

  1. Basic Algebra I, II: Nathan Jacobson.
  2. Algebra: Serge Lang.
  3. Abstract Algebra: David S. Dummit, Richard M. Foote.
  4. Algebra: Michael Artin.

Topics covered:

Lecture 1 (Aug 6): monoids, examples, homomorphism, isomorphism, monoids and groups of transformations.

Lecture 2 (Aug 8): congruences on monoids, the quotient monoid, universal property, the first isomorphism theorem for groups and monoids, free monoids.

Lecture 3 (Aug 10): free groups, universal property, construction.

Assignment 1

Lecture 4 (Aug 13): constructing groups using generators and relations.

Lecture 5 (Aug 20): group actions, examples, new actions from old.

Lecture 6 (Aug 24): orbits, stabilizers, actions on left coset spaces, the class equation.

Assignment 2

Lecture 7 (Aug 27): applications of group actions, the sylow theorems.

Lecture 8 (Aug 29): burnside's lemma, applications.

Lecture 9 (Aug 31): direct and semidirect products of groups.

Assignment 3

Midterm 1

Lecture 10 (Sep 3): linear functionals, the dual space, bilinear forms, matrix of a form, change of basis, equivalence (isomorphism) of forms.

Lecture 11 (Sep 5): left and right radicals of a form, nondegeneracy, the left and right adjoints of a linear operator relative to a nondegenerate form.

Lecture 12 (Sep 12): symmetric and alternating forms, consequences of nondegeneracy, subspaces and perpendiculars.

Assignment 4

Lecture 13 (Sep 14): finding "nice" bases for alternating and symmetric forms, sylvester's theorem.

Lecture 14 (Sep 17): positive definite forms on real vector spaces, hermitian forms on complex vector spaces.

Lecture 15 (Sep 19): spectral theorem for self-adjoint operators on real and complex vector spaces, another proof of sylvester's theorem.

Notes on Linear Algebra, by V.S.Sunder (suggested reference for the Spectral theorem)

Assignment 5

Lecture 16 (Sep 21): rings, ideals, homomorphisms, domains, field of fractions of a domain.

Lecture 17 (Sep 26): polynomial rings, universal property, the monoid algebra over a commutative ring.

Lecture 18 (Sep 28): division algorithm, pids, divisibility.

Lecture 19 (Oct 1): irreducibles, primes, ufd, characterization of a ufd.

Lecture 20 (Oct 3): gauss' lemma, a polynomial ring over a ufd is a ufd.

Assignment 6

Midterm 2

Lecture 21 (Oct 12): modules, examples, submodules, quotients, first isomorphism theorem.

Lecture 22 (Oct 15): bimodules, the space of morphisms between bimodules, free modules of finite rank, morphisms between free modules, matrices.

Lecture 23 (Oct 17): matrices over commutative rings, determinants, invertibility, change of base formula.

Assignment 7

Lecture 24 (Oct 22): normal form of a matrix over a pid: existence.

Lecture 25 (Oct 29): uniqueness of normal form, modules over pids, cyclic modules.

Lecture 26 (Oct 31): structure theorem for finitely generated modules over pids.

Lecture 27 (Nov 5): torsion submodule, primary components.

Lecture 28 (Nov 7): uniqueness of elementary divisors and invariant factors.

Lecture 29 (Nov 9): application to finitely generated abelian groups and linear operators on finite dimensional vector spaces, the rational canonical form.

Assignment 8

Assignment 9

Lecture 30 (Nov 12): the tensor product of modules, definition, universal property, balanced maps.

Lecture 31 (Nov 14): functoriality, direct sums, bimodules.

Lecture 32 (Nov 16): associativity of tensor products, n-fold tensor products, tensor products of vector spaces.

Lecture 33 (Nov 19): algebras, tensor product of algebras, the tensor algebra.

Video (tensor products of vector spaces).

Assignment 10

Lecture 34 (Nov 21): exterior and symmetric algebras, categories, definition and examples.

Lecture 35 (Nov 23): functors, covariant, contravariant and bifunctors, universal objects, adjoint functors.

Assignment 11