Updated a link in README.md
lie-algebras-and-their-representations
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
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
Changed the title of chapter 3
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
Changed the notation for the Jordan decomposition
Merge branch 'master' into font-change
Changed the styling of a weight diagram
Merge branch 'master' into font-change
Added a link to the compiled document in the README
Merge branch 'master' into font-change
Simplfied a URL
Changed the color of links
Replaced the monospace font for the TXTT font
Fixed a typo
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