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Semi-simple Lie algebras are determined by their Iwasawa subalgebras

Jonathan Epstein, Michael Jablonski

Abstract

Using tools from the geometry of Einstein solvmanifolds, we give a geometric argument that a semi-simple Lie algebra (of non-compact type) is completely determined by its Iwasawa subalgebra. Furthermore, we produce an algebraic procedure for recovering the semi-simple (of non-compact type) from its Iwasawa subalgebra.

Semi-simple Lie algebras are determined by their Iwasawa subalgebras

Abstract

Using tools from the geometry of Einstein solvmanifolds, we give a geometric argument that a semi-simple Lie algebra (of non-compact type) is completely determined by its Iwasawa subalgebra. Furthermore, we produce an algebraic procedure for recovering the semi-simple (of non-compact type) from its Iwasawa subalgebra.
Paper Structure (5 sections, 3 theorems, 16 equations, 1 figure)

This paper contains 5 sections, 3 theorems, 16 equations, 1 figure.

Table of Contents

  1. Proof of Theorem \ref{['thm: iwasawa determines its semi-simple']}
  2. Construction of semi-simple of non-compact type from the Iwasawa
  3. Building $\mathfrak g$ from $\mathfrak g_{\geq 0}$.
  4. Recovering $\mathfrak m$ from $\mathfrak s$.
  5. Nilradical

Key Result

Theorem 1

Consider two, real semi-simple Lie algebras $\mathfrak g_1$ and $\mathfrak g_2$ of non-compact type with corresponding Iwasawa subalgebras $\mathfrak s_1$ and $\mathfrak s_2$. If $\mathfrak s_1$ and $\mathfrak s_2$ are isomorphic, then $\mathfrak g_1$ is isomorphic to $\mathfrak g_2$. Moreover, ever

Figures (1)

  • Figure 1: The Satake diagram of EIII.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 2
  • Remark
  • Lemma 3
  • proof
  • proof : Proof of Lemma \ref{['lem: compact derivations of iwasawa']}
  • proof : Proof of Theorem \ref{['thm: iwasawa determines its semi-simple']}, cont.
  • Example 4