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Course11110852051 lectures

Representation Theory of General Linear Lie Algebra

Prof. R. Venkatesh · IISc Bangalore

Start with lecture 1 At NPTEL

Lectures

week-01

  1. Lec 01 Basic theory of Lie algebras. Video
  2. Lec 02 Basic theory of Lie algebras (Continued). Video
  3. Lec 03 Basic theory of Lie algebras (Continued). Video
  4. Lec 04 Representation theory of Lie algebras. Video
  5. Lec 05 Representation theory of Lie algebras (Continued). Video

week-02

  1. Lec 06 Representation of Lie algebras. Video
  2. Lec 07 Classification of 2 dimensional representation of two dimensional non-abelian Lie algebra. Video
  3. Lec 08 Isomorphism Theorem. Video
  4. Lec 09 Characterization of completely reducible module. Video
  5. Lec 10 Representation theory of sl2(C). Video

week-03

  1. Lec 11 Irreducible representation of sl2(C) Video
  2. Lec 12 Irreducible representation of sl2(C) (Continued). Video
  3. Lec 13 Complete reducibility of sl2(C). Video
  4. Lec 14 Complete reducibility of sl2(C) (continued). Video
  5. Lec 15 Representation theory of sl2(C)-basic observation Video
  6. Lec 16 Application of sl2(C) representation theory in combinatorics. Video

week-04

  1. Lec 17 Constructions of new representations. Video
  2. Lec 18 Construction of universal algebras I. Video
  3. Lec 19 Construction of universal algebras II Video
  4. Lec 20 Non-degenerate and invariant bilinear forms. Video
  5. Lec 21 Schur’s lemma and Killing form Video

week-05

  1. Lec 22 Killing form of general and special linear Lie algebras. Video
  2. Lec 23 The universal enveloping algebra. Video
  3. Lec 24 Properties of the universal enveloping algebra Video
  4. Lec 25 The Casimir operator for representations of gln(C) and sln(C). Video

week-06

  1. Lec 26 Weyl’s theorem of complete reducibility Video
  2. Lec 27 The structure of sln+1(C). Video
  3. Lec 28 Representations of sln+1(C) I. Video
  4. Lec 29 Representations of sln+1(C) II. Video

week-07

  1. Lec 30 Representations of sln+1(C) III. Video
  2. Lec 31 Casimir operator and highest weight modules I. Video
  3. Lec 32 Casimir operator and highest weight modules II. Video
  4. Lec 33 Irreducible representations of sln+1(C). Video
  5. Lec 34 Verma module and its irreducible quotient. Video

week-08

  1. Lec 35 Verma modules and standard cyclic irreducible modules. Video
  2. Lec 36 Finite dimensional standard cyclic irreducible modules. Video
  3. Lec 37 Standard cyclic irreducible modules. Video
  4. Lec 38 Character theory. Video

week-09

  1. Lec 39 Character theory - II Video
  2. Lec 40 Deniminator Identity - Prep 1 Video
  3. Lec 41 Deniminator Identity - Prep 2 Video
  4. Lec 42 Proof of the denominator identity Video

week-10

  1. Lec 43 Freudenthal's formula Video
  2. Lec 44 The Weyl character formula Video
  3. Lec 45 The Laplacian operator and the Weyl Character Formula Video
  4. Lec 46 Proof of the Weyl Character Formula Video

week-11

  1. Lec 47 Applications of Weyl Character formula Video
  2. Lec 48 The Weyl dimension formula, Schur polynomials Video

week-12

  1. Lec 49 Semi-standard Young Tableaux Video
  2. Lec 50 BGG resolution Video

Live Session

  1. Live Session 04-10-2024 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISc Bangalore
Department
Mathematics
Teacher
Prof. R. Venkatesh
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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