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Automorphism groups and derivation algebras of Hamiltonian Lie algebras

Pradeep Bisht, Suman Rani, Santanu Tantubay

Abstract

In this paper, we compute the automorphism group and derivation algebra of the Hamiltonian Lie algebra $\mathcal{H}_{N}$ and its derived subalgebra $\mathcal{H}_{N}'$, where $N$ is an even positive integer. The automorphism groups are shown to be $\mathbf{GSp}_{N}(\mathbb{Z})\ltimes (\mathbb{\mathbb{K}}^{\times})^{N}$ for both Lie algebras and we prove that all derivations are inner for the Hamiltonian Lie algebra, also we compute the full derivation space for the derived subalgebra of Hamiltonian Lie algebra.

Automorphism groups and derivation algebras of Hamiltonian Lie algebras

Abstract

In this paper, we compute the automorphism group and derivation algebra of the Hamiltonian Lie algebra and its derived subalgebra , where is an even positive integer. The automorphism groups are shown to be for both Lie algebras and we prove that all derivations are inner for the Hamiltonian Lie algebra, also we compute the full derivation space for the derived subalgebra of Hamiltonian Lie algebra.
Paper Structure (4 sections, 19 theorems, 105 equations)

This paper contains 4 sections, 19 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Automorphism groups of Hamiltonian Lie algberas
  4. Derivations of Hamiltonian Lie algebra

Key Result

Theorem 1

Let $N\geq 2$ be even, then $\operatorname{Aut}(\mathcal{H}_N)\cong \operatorname{Aut} (\mathcal{H}_N^\prime)\cong \mathbf{GSp}_N(\mathbb{Z})\ltimes (\mathbb{K}^{\times})^{N}$, where $\mathbf{GSp}(\mathbb{Z})$ is the conformal symplectic group over $\mathbb{Z}$, consisting of matrices $\mathbf{Q}\in

Theorems & Definitions (34)

  • Theorem : A
  • Theorem : B
  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • ...and 24 more