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On $(α,1,0)$-derivations of anti-commutative algebras

Edison Alberto Fernández-Culma

Abstract

We solve an open problem concerning the well-known $(α,β,γ)$-derivations, proving that the spaces of $(α,1,0)$-derivations of any Lie algebra are isomorphic ($α\neq 0,1$). Also, we prove sharp bounds for the invariants functions defined by such spaces.

On $(α,1,0)$-derivations of anti-commutative algebras

Abstract

We solve an open problem concerning the well-known -derivations, proving that the spaces of -derivations of any Lie algebra are isomorphic (). Also, we prove sharp bounds for the invariants functions defined by such spaces.
Paper Structure (4 sections, 9 theorems, 20 equations)

This paper contains 4 sections, 9 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The family $\{\phi_{n,t}\}_{t \in \mathbb{K}}$ is infinite
  3. The family $\{\widehat{\phi}_{n,t}\}_{t \in \mathbb{K}}$ is finite and has three elements
  4. Upper and lower bounds of $\widehat{\phi}_{n,t}$, $t\neq 0,1$

Key Result

Proposition 2.1

Let $\mathfrak{A}_{s}=(V,\mu_{s})$ be the one-parameter family of four-dimensional anti-commutative algebras given by with $s \in \mathbb{K}$. If $t\neq 0,1$, then the value of the invariant function $\phi_{4,t}$ at $\mathfrak{A}_{s}$ is and therefore $\phi_{4,t_1} = \phi_{4,t_2}$ if and only if $t_1 = t_2$ (with $t_1, t_2 \neq 0,1$).

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2: NovotnyHrivnak
  • Proposition 2.1
  • Definition 2.1: Centralizer and Derived algebra
  • Remark 2.1
  • Remark 2.2
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 18 more