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A Course on Lie algebras and Chevalley groups

Meinolf Geck

TL;DR

Geck's notes provide a self-contained path from foundational Lie-algebra concepts to semisimple theory and the CK-type framework, with an emphasis on modern Lusztig canonical-basis perspectives. The text integrates algebraic constructions with weight- and root-space ideas, Weyl-group symmetry, and explicit structure constants, while foregrounding algorithmic and computational approaches to examples. It lays the groundwork for Chevalley groups by deriving Cartan decompositions and simple-ideal decompositions from purely algebraic data, connecting to both classical types like ${\mathfrak{sl}}_n$ and broader CK-type algebras. The work aims to bridge classical treatments (Steinberg, Carter) with canonical-basis methods and to motivate a broader book-length development of Chevalley groups and their representations.

Abstract

These are expanded notes from graduate courses about Lie algebras and Chevalley groups held at the University of Stuttgart. In the 1950s Chevalley showed how linear groups over arbitrary fields could be obtained~ -- ~by a uniform procedure~ -- ~from the simple Lie algebras over $\C$ occurring in the Cartan--Killing classification. Together with subsequent variations, Chevalley's work had a profound and long-lasting impact on group theory and Lie theory in general. Classical, and widely used references are the lectures notes by Steinberg (1967) and the monograph by Carter (1972). Our aim here is to present a self-contained introduction to the theory of Chevalley groups, based on recent simplifications arising from Lusztig's fundamental theory of ``canonical bases''. A further feature of our text is that we explicitly incorporate algorithmic methods in our treatment, both for the handling of substantial examples and regarding some aspects of the general theory. Eventually, this may turn into a book project.

A Course on Lie algebras and Chevalley groups

TL;DR

Geck's notes provide a self-contained path from foundational Lie-algebra concepts to semisimple theory and the CK-type framework, with an emphasis on modern Lusztig canonical-basis perspectives. The text integrates algebraic constructions with weight- and root-space ideas, Weyl-group symmetry, and explicit structure constants, while foregrounding algorithmic and computational approaches to examples. It lays the groundwork for Chevalley groups by deriving Cartan decompositions and simple-ideal decompositions from purely algebraic data, connecting to both classical types like and broader CK-type algebras. The work aims to bridge classical treatments (Steinberg, Carter) with canonical-basis methods and to motivate a broader book-length development of Chevalley groups and their representations.

Abstract

These are expanded notes from graduate courses about Lie algebras and Chevalley groups held at the University of Stuttgart. In the 1950s Chevalley showed how linear groups over arbitrary fields could be obtained~ -- ~by a uniform procedure~ -- ~from the simple Lie algebras over occurring in the Cartan--Killing classification. Together with subsequent variations, Chevalley's work had a profound and long-lasting impact on group theory and Lie theory in general. Classical, and widely used references are the lectures notes by Steinberg (1967) and the monograph by Carter (1972). Our aim here is to present a self-contained introduction to the theory of Chevalley groups, based on recent simplifications arising from Lusztig's fundamental theory of ``canonical bases''. A further feature of our text is that we explicitly incorporate algorithmic methods in our treatment, both for the handling of substantial examples and regarding some aspects of the general theory. Eventually, this may turn into a book project.
Paper Structure (34 sections, 151 theorems, 486 equations, 16 tables)

This paper contains 34 sections, 151 theorems, 486 equations, 16 tables.

Table of Contents

  1. Introducing Lie algebras
  2. Non-associative algebras
  3. Matrix Lie algebras and derivations
  4. Solvable and semisimple Lie algebras
  5. Representations of Lie algebras
  6. Lie's Theorem
  7. The classical Lie algebras
  8. Semisimple Lie algebras
  9. Weights and weight spaces
  10. Lie algebras of Cartan--Killing type
  11. The Weyl group
  12. Semisimplicity
  13. Classical Lie algebras revisited
  14. The structure constants $N_{\alpha,\beta}$
  15. Lusztig's canonical basis
  16. ...and 19 more sections

Key Result

Lemma 1.1.2

Let $n\geqslant 1$ and $\chi_1,\ldots,\chi_n\colon M\rightarrow k$ be distinct characters. Then $\chi_1,\ldots,\chi_n$ are linearly independent (in the vector space of all functions $f\colon M\rightarrow k$).

Theorems & Definitions (338)

  • Example 1.1.1
  • Lemma 1.1.2: Linear independence of characters
  • Example 1.1.3
  • Remark 1.1.4
  • Lemma 1.1.5
  • Definition 1.1.6
  • Example 1.1.7
  • Example 1.1.8
  • Example 1.1.10
  • Definition 1.1.11
  • ...and 328 more