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Classification of Certain Regular Subalgebras of $\mathfrak{sl}(n, \mathbb C)$ up to Conjugacy

Shreya Dhar

Abstract

In this paper we will be classifying some regular upper-triangular subalgebras of $\mathfrak{sl}(n, \mathbb C)$ up to conjugacy by matrices in $SL(n, \mathbb C)$. We do so for dimension 2, codimension 1, and codimension 2 subalgebras. We prove some general results for codimension $k$. The approach we use reduces an abstract classification problem to a combinatorial one, which we solve through a mixture of inductive and computational approaches.

Classification of Certain Regular Subalgebras of $\mathfrak{sl}(n, \mathbb C)$ up to Conjugacy

Abstract

In this paper we will be classifying some regular upper-triangular subalgebras of up to conjugacy by matrices in . We do so for dimension 2, codimension 1, and codimension 2 subalgebras. We prove some general results for codimension . The approach we use reduces an abstract classification problem to a combinatorial one, which we solve through a mixture of inductive and computational approaches.

Paper Structure

This paper contains 15 sections, 51 theorems, 25 equations, 5 tables.

Table of Contents

  1. Notation
  2. General Theorems, Lemmas, and Definitions
  3. Lie Bracket
  4. Lemmas on closure under Lie bracket:
  5. Lemmas on Invariance
  6. Lemmas on Conjugacy
  7. Adjoint Action of $H_i$
  8. Adjoint Action of $H_{pq}$ on $\text{span }(E)$
  9. Adjoint Action of $H_{pq}$ on $\text{span }(E \setminus \{ E_{i,i+1}) \}$
  10. Some Classifications for $\mathfrak{sl}(n, \mathbb{C})$
  11. Classification of $\frac{n(n+1)}{2} - 1$ dimensional upper-triangular regular subalgebras
  12. Classification of Codimension 2 subalgebras
  13. Classification of 2 Dimensional regular Algebras
  14. Some More General Results
  15. Acknowledgements

Key Result

Lemma 1

(Nilpotent Element Removal) Fix $i < j$. Let $L$ be a upper-triangular regular subalgebra with the standard basis, without $E_{ij}$ as a basis element. Then for $k \geq i + 1$ and $k \leq j - 1$, $E_{ik}$ is not a basis element in the subalgebra or $E_{kj}$ is not a basis element in the subalgebra.

Theorems & Definitions (76)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Corollary 1
  • Lemma 2
  • Lemma 3
  • Corollary 2
  • Corollary 3
  • Definition 3
  • ...and 66 more