lie-algebras-and-their-representations

Updated the references to my academic website

Updated the README

Changed the document title

Added an "under construction" note to the preface

Broke up a paragraph in the acknowledgments

Added a link to a copy of the CC4 license

Restructured the preface

Minor tweak in language

Added some TODO items

Minor tweak in notation

Started to work on hydrating the final chapter

Fixed some typos

Added a TODO item

Added a proof

Added a proof

Minor tweak

Added some comments on the graph structure of 𝓑

Added some comments on the proof of a proposition

Added the classification of coherent sl(2)-families as an example

Demoted a proposition to a lemma

Revised some TODO items

Minor tweak to the statement of a lemma

Fixed a typo

Added a joke 😛

Fixed a calculation error

Tweaked the statement of a theorem

Changed the spacing of the diagrams on the connected components of 𝓑

Added some proofs to the last chapter

Fixed multiple issues in the discussion of the graph 𝓑

Added the statement of a lemma

Demoted a proposition to a lemma

Changed the notation for the set of parameters for coherent families

Added further details to a theorem

Added a definition

Added sp(2n) to the list of simple Lie algebras

Revised the comments on regular and integral sl(n)-sequences

Standardized the notation for the dot action

Added further results on Weyl groups

Fixed two typos

Minor tweak in notation

Changed the notation of the last chapter

Changed the notation for the graph 𝓟

Removed a TODO item

Added a TODO item

Changed the notation for the elements of the defining basis of Δ for sl(3)

Elaborated on a TODO item

Fixed a typo

Added a skeleton for the chapter on coherent families

Elaborated on the definition of the symplectic Lie algebras

Properly finished off the chapter on simple weight modules

Added a file for the chapter on the classification of coherent families

Fixed a typo

Minor tweak in notation

Added a "last updated" notice to the licesing information

Minor tweaks to the intro to the section on Verma modules

Tweaker the text in the preface

Minor tweak in notation

Fixed a small typo

Fixed a typo

Revised the discussion on Fernando's work

Corrected the remarks on Verma modules of antidominant weights

Corrected the definition of the order in the root lattice

Restructured the section on Verma modules

Fixed a typo

Removed duplicate labels

Renamed some files

Tweaked some notation

Fixed another typo

Fixed a typo

Turned a footnote into a regular sentence

Simplified an example

Corrected the last chapter

Removed the catalographic card from the notes

Minor tweak in notation

Changed the notation for the PBW Theorem

Added further comments on simple modules of the direct sum

Simplified the code for boxed theorems

Added a proof of a proposition

Cleaned the code for boxed theorem environments

Removed manual page breaks

Further refined the styles of boxed theorem environments

Made it so that enumerate items are colored

Refined the style of the theorems

First draft of an implementation of colored/boxed theorem environments

Replaced the term "admissible" with the term "bounded"

Reoriented communative diagrams to avoid arrows going up

Fixed a typo

Clarified a sentence

Added a clarification to the discussion on cohomology

Added a clarification to the discussion on cohomology

Fixed the name of the standard resolution of the trivial module

Fixed a typo

Replaced a reference

Fixed a major error on cohomology

Clarified the analysis of 𝔰𝔩₂

Minor improvement in the notation of a proof

Fixed a minor typo

Added some qualifiers to the statement about the Lie functor for algebraic groups

Minor improvement in notation

Minor tweak in language

Fixed a few typos

Minor tweak to an equation

Corrected errors in a reference

Added further comments to the section on cohomology

Added funther comments on algebraic groups

Minor tweak in notation

Cleaned the section on Verma modules

Removed the TODO list

Tweaked some sentences

Added further comments on the proof of the Highest Weight Theorem

Fixed a comment on the monoidal structure of g-Mod

Repharased a sentence

Added further details to a theorem

Fixed a typo

Added further details to a remark

Changed the notation for the Casimir element

Added some further comments on the Chevalley-Eilenberg resolution

Changed the notation for the Killing form

Updated the TODO list

Changed the notation for twisted modules (yet again)

Minor correction to the introduction

Fixed some typos

Extracted a result to a couple of examples

Credited Weyl in a proof

Changed the notation for differential operators

Fixed a typo

Demoted a theorem to a proposition

Fixed a typo

Fixed some typos

Changed the notation for twisted modules

Added some further clarification on what we assume

Fixed a typo

Added an index to the document

Updated the .gitignore

Cleaned the code for commutative diagrams

Added a catalographic card and a title page

Added a TODO list

Fixed some typos

Switched the notation for a dominat weight in chapter 1

Added further clarifications on invariants

Fixed some typos

Switched the notation for completely reducible moddules

Fixed a typo

Fine-tuned the notation for Methieu's coherent extension

Changed the notation for the Weyl group

Fixed some typos

Switched the notation for a dominat weight in chapter 5

Switched the notation for a dominat weight in chapter 4

Reworked the introduction of the 3rd chapter

Merge branch 'master' into module-notation

Switched the notation for a dominat weight in chapter 3

Switched the notation for modules/representations in the third chapter

Switched the notation for modules/representations in the second chapter

Added a clarification

Changed the notation for Mathieu's coherent extension

Fixed the reference to Kirillov

Fixed some typos

Fixed a major error in the proof of complete reducibility

Minor tweak in the references

Clarified the ortogonality relations in the weight diagrams of sl3

Repharased a sentence

Added a reference to the concept of adjoint functors

Added clarifications on the functoriality of a construction

Minor tweak in language

Added a reference to the non-fullyness of the Lie functor for algebraic groups

Renumbered theorems and definitions

Fixed some typos

Repharased a sentence

Changed a single word

Fixed some typos

Repharased a sentence

Fixed some typos

Removed a reference to the title of a book

Clarified the PBW theorem

Fixed a typo

Added the definition of the dimension of a Lie algebra

Fixed multiple typos

Capitalized named theorems and definitions

Sorted the bibliography

Removed abreviations

Changed the notation in an example

Fixed a typo

Merge branch 'master' into font-change

Merge branch 'master' into font-change

Merge branch 'master' into font-change

Merge branch 'master' into font-change

Changed the color of links

Replaced the monospace font for the TXTT font

Replaced the standard LaTeX font with the Palladio font family

Changed the notation for the localization of a ring

Minor tweak in notation

Minor tweak in language

Minor tweak in notation

Fixed a typo in the references

Added some further thanking

Fixed a typo

Added a minor clarification

Added a drawing to the section on sl2

Fixed multiple typos

Made it so that the pages beforen the start of the first chapter are not numbered

Added a blank page after the dedication

Fixed multiple typos

Fixed typos

Made the "1" in "1-dimensional" upwrite

Revised the forth chapter

Revised the third chapter for a third time!

Minor clarification

Fixed a definition

Cleaned the references

Clarified some notation

Simplified the preamble

Removed a redefinition of a macro

Fixed a typo

Removed the macros for double-struck letters

Incorporated hidden statements into the statement of a theorem

Added a proof of a technical result

Removed duplicate words

Fixed some typos

Fixed a typo

Fixed multiple typos

Removed Lucas full name

Revised the final chapter

Revised the forth chapter

Clarified the action of g in V*

Added some remarks

Replaced the word "killed" by "annihilated"

Revised the third chapter (again)

Fixed a major typo

Revised the second chapter (again)

Revised the first chapter (again)

Revised the third chapter

Fixed a typo

Added notes on how coherent families are useful in other problems

Automated the "47 pages left" thing from chapter 2

Revised the second chapter

Standardized the word "one-dimensional"/"1-dimensional"

Standardized the word "nonzero"/"non-zero"

Revised the first chapter

Fixed multiple typos

Minor styling tweak

Fixed the alignment of the title in the cover

Clarified some notation

Rewrote the preface

Changed the title of the introduction

Fixed a TODO item

Fixed a TODO item

Completed the notes with a catchphrase

Added a small conclusion to the introduction

Added some remarks

Removed a TODO item

Added the direct sum of Lie algebras to the list of examples

Added the box product of representations to the list of examples

Reworked some of the discussion on representations

Added a conclusion to the chapter on Mathieu's work

Added the definition of the subrepresentation generated by an element

Added a small clarification in a proof

Added a TODO item

Added notes on the "uniqueness" part of the highest weight theorem

Promoted some TODO items

Added notes on the intuition behind a proof

Tied up transitions between sections in chapter 4

Fixed an error

Fixed some typos and added a clarification

Changed the notation for the height of a weight

Fixed a typo

Reworked the subdivision of the forth chapter

Fixed a sign in some equations

Added a TODO item

Defined the action of the Weyl group in g

Changed the notation for elements of the Weyl group

Removed TODO items

Added a TODO item

Reordered the contents of the chapter on the general finite-dimensional setting

Added TODO items

Rewrote the introduction of the chapter on sl2 and sl3

Changed the names of some sections

Reordered the contents of the section on sl3

Minor improvement in the notation of a proof

Added TODO items

Fixed the proof on the theorem on D-modules

Removed unnecessary comment

Modified the proof of the dominant integral weight theorem for sl2

Reordered some equations

Incorporated the section on sl3 in the introduction of the chapter on sl3

Removed a stupid epigraph

Fixed a typo

Reordered some equations

Moved the discussion on representations of arbitrary semisimple algebras to a separate chapter

Tied up the conclusion of the chapter on complete reducibility

Cleaned TODO items

Added notes on the proof of Cartan's criterium for semisimplicity

Moved the discussion on sl2 and sl3 to a new chapter

Started to work on the section on invariant bilinear forms

Incorporated much of the discussion on semisimplicity and complete reducibility to the chapter on sl2

Renamed a file

Renamed the second chapter

Added a TODO item

Added the center of a Lie algebra to the list of examples of Abelian algebras

Added the trivial representation to the list of examples

Removed an example

Fixed multiple typos

Fixed some typos and clarified some things

Added the regular module to the list of examples of representations

Fixed a TODO item

Cleaned TODO items

Added a remark on tensor products

Clarified some notation

Added TODO items

Hydrated the section on representations

Reworked a sentence

Fixed some typos

Fixed an example

Added a TODO item

Added a brief proof of Frobenius reciprocity

Updated the TODO list

Hydrated the section the basics of Lie algebras

Reordered the contents of the introduction

Fixed a typo

Started to hydrate the introductory chapter

Cleaned the introduction of the second chapter

Clarified a comment on Lie algebra cohomology

Edited the example of the Lie algebra of an algebraic group

Fixed a typo

Fixed a typo

Added proofs to some of the theorems of the introduction

Repharased the PBW theorem a bit

Changed the notation for the algebra of derivations

Finished the blueprint of the introduction

Added the Lie algebra of an algebraic group to the list of examples

Added the adjoint representation to the list of examples of representations

Moved the discussion on representations to after the basic results/definitions of Lie algebras

Started to work on the blueprint for the introduction

Promoved some TODO items

Defined the notation for the components of a g-module

Added an adjective

Updated the TODO list

Removed an unnecessary file

Clarified the meaning of "Zariski topology" in the second chapter

Removed some footnotes

Added references to missing proofs

Fixed some typos

Added comments on the lack of functoriality of some of the constructions in the last chapter

Added comments on the fact that some automorphisms factor trought some maps

Moved the proposition on the multiplicity of irreducible weight modules inside completely reducible weight modules

Edited some TODO items

Hydrated the section on the twisted localization construction

Removed an unnecessary theorem

Finished hydrating the section on the basics of coherent families and coherent extensions

Added an alternative definition of a simple coherent family and some comments on the equivalence between the definitions

Changed the notation for irreducible/simple objects in the category of coherent faimlies

Wrote the section on the motivation behind coherent families

Added a footnote on the origins of notation

Minor fixes

Reworked the section on Fernando's work

Hydrated the section on Fernando's work

Fixed a typo

Clarified a definition

Minor tweak in language

Formatted references.bib

Tied up the example of a representation which is not a weight module

Started to work on hydrating the final chapter

Added a footnote

Added a section label

Removed an unused equivalence from a proposition

Edited some TODO items

Added the example of the sl2-module on Laurent polynomials

Corrected the statement of Fernando's theorem

Edited some TODO items

Fixed multiple typos

Removed a reduntant formula

Minor tweaks

Minor correction

Finished the proof of a technical lemma

Fixed a typo

Added a TODO item

Tweaked the formating of a commutative diagram

Minor tweak

Fixed a typo

Fixed a minor error

Mostly finished the proof of Mathieu's classification of which submodules of an irreducible coherent family are cuspidal

Updated some TODO items

Changed the notation for the centralizer of the Cartan subalgebra in the universal enveloping algebra

Proved the uniqueness of Mathieu's Ext extension

Added TODO items

Added a proof the existence of the torsion automorphisms

Moved the proof that an admissible module is contained in any of its semisimple coherent extensions to a separate theorem

Formulated the theorem on the semisimplification of a cohenrent family in more abstract terms

Replaced the term semisimple for the term completely reducible

Minor corrections

Fixed some typos

Removed an unnecessary whitespace

Fixed a minor error in the definition of a coherent extension

Added the results we need on Ore's localization

Changed the notation for the localization of the modules

Added a proof of the fact that the localization of an admissible module is well behaved

Fixed a typo

Changed the notation for elements of a basis

Moved the discussion on the existence of a coherent extension to a separate theorem

Fixed a typo

Added a TODO item

Added references to the discussion on Lie algebra cohomology on "Symplectic techniques in physics"

Changed the notation for the Weyl group

Fixed a typo

Incorporated the discussion on how cuspital modules fits inside their extensions to the proof of the existance of Ext

Changed the notation for the twisting isomorphisms

Started to work on the proof of the existence of Ext

Added lemmas necessary to prove the existance of Mathieu's Ext extension

Moved the part of the theorem on Ext about the central characters of the irreducible submodules of Ext to a separate theorem

Moved the theorem on the existence of good coherent extensions to the end of the text

Added TODO items

Added a TODO item

Fixed a typo

Added further compilation instructions to the README

Added TODO items

Demoted a proposition to the mere status of a lemma

Added minor clarifications

Fixed some typos

Added further clarifications to the proof that V fits nicely inside Ext(V)

Fixed a typo

Fixed the notation for the composition series in the proof that the semisimplification is a coherent family

Fixed the proof that cuspidal modules fit nicely inside their Exts

Fixed the definition of the semisimplification of a coherent family

Added a description of the weight spaces of a subrepresentation of a weight module

Added a proof of the fact that quotients of weight modules are weight modules

Minor change in notation

Removed unnecessary whitespace

Added the definition of the semisimplification of a coherent family

Added a proof of the fact that any irreducible cuspidal representation is given by M[lambda] for some coherent family lambda

Fixed a typo

Updated the TODO list

Added proper references to the chapter on Mathieu's work to the chapter on semisimple Lie algebras

Removed unnecessary whitespace

Fixed typos and removed unnecessary whitespace

Completed the proof of complete reducibility

Completed the proof that every finite-dimensional representation of a semisimple Lie algebra is a weight module

Added some items to the TODO list

Started to work on the chapter about Mathieu's paper

Added a TODO list

Fixed the comment on the canonical acyclic resolution of the trivial representation

Fixed some typos

Added some clarification on language

Fixed an error in an equation

Added an example of Verma module

Clarified a corolarry on Lie algebra cohomology

Properly formated multiline exact sequences

Added information on how to compile the document to the README

Adicionados um README e uma licensa

Adicionado comentário sobre o fato de que nem todo módulo de dimensão infinita é módulo de peso

Revisão ortrográfica

Terminei de trabalhar na prova do teorema do peso integral dominante

Corrigido o comentário sobre a prova de redutibilidade completa do Weyl

Comecei a trabalhar na prova do teorema do peso integral dominante

Removidos espalos desnecessários

Ajustados comentários iniciais sobre a prova do teorema do peso integral dominante

Ajustados os comentários sobre diagonalização simultânea

Corrigida a prova de que todo módulo de dimensão finita é módulo de peso

Pequena quebra em um parágrafo mais longo

Concertados diversos probleminhas ao longo do primeiro capítulo

Fixed the discussion on the maximal weight

Pequenos ajustes gerais no primeiro capítulo

Concertados mais alguns errinhos gramaticais e frases estranhas

Concertados varios typos e removidos espaços desnecessários

Terminada (em larga medida) a seção sobre a prova de redutibilidade completa

Comecei a trabalhar na prova de redutibilidade completa

Concertado o plural no primeirocapítulo

Adicionada uma definição mais precisa da subálgebra de Cartan

Ajustado o modelo de numeração dos teoremas

Corrigidos alguns pecados latéticos

Comecei a trabalhar na introdução do capítulo sobre álgebras semisimples

Corrigida a formatação do documento

Alterada a notação para a forma de Killing, de modo a evitar conflitos com a notação do corpo base

Renomeado um arquivo

Limpado um pouco do preâmbulo

Removidas as dependências nos meus pacotes pessoais

Alterado o estilo da capa

Alterada a notação do corpo base

Ajustada a dedicatória, o título e o prefácio

Limpadas mais algumas patologias geometricas e adicionados TODOs

Limpadas algumas das patologias geometricas das quais padecia esse capítulo

Removidas images no formato PDF do arquivo ZIP do código fonte

Adicionadas mais extensões ao .gitignore

Commit inicial