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Local superderivation and super-biderivation on generalized quaternion algebra

Hassan Oubba

TL;DR

This work analyzes local and 2-local superderivations and super-biderivations on the generalized quaternion algebra $\mathcal{H}^{a,b}$ viewed as a Lie superalgebra. It proves that every local superderivation on $\mathcal{H}^{a,b}$ is a superderivation, and provides a complete degree-wise characterization of super-biderivations. Specifically, degree-0 super-biderivations admit an explicit form parameterized by $\lambda\in\mathcal{R}$, while degree-1 super-biderivations vanish. Consequently, $\mathrm{Der}_s(\mathcal{H}^{a,b})=\mathrm{Inn}_s(\mathcal{H}^{a,b})$ and $\mathrm{Out}_s(\mathcal{H}^{a,b})=0$, clarifying the derivation structure and its inner/outer decomposition for these algebras.

Abstract

Let $\mathcal{H}^{a,b}$ be the generalized quaternion algebra over a unitary commutative ring. This paper aims to investigate super-biderivations and local superderivations on the generalized quaternion algebra, which is viewed as a class of Lie superalgebra. It turns out that on generalized quaternion algebras, any local superderivation is a superderivation.

Local superderivation and super-biderivation on generalized quaternion algebra

TL;DR

This work analyzes local and 2-local superderivations and super-biderivations on the generalized quaternion algebra viewed as a Lie superalgebra. It proves that every local superderivation on is a superderivation, and provides a complete degree-wise characterization of super-biderivations. Specifically, degree-0 super-biderivations admit an explicit form parameterized by , while degree-1 super-biderivations vanish. Consequently, and , clarifying the derivation structure and its inner/outer decomposition for these algebras.

Abstract

Let be the generalized quaternion algebra over a unitary commutative ring. This paper aims to investigate super-biderivations and local superderivations on the generalized quaternion algebra, which is viewed as a class of Lie superalgebra. It turns out that on generalized quaternion algebras, any local superderivation is a superderivation.

Paper Structure

This paper contains 6 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local and 2-local superderivations on $\mathcal{H}^{a,b}$
  4. super-biderivation on $\mathcal{H}^{a,b}$
  5. Characterization of $BDer_0(\mathcal{H}^{a,b})$
  6. Characterization of $BDer_1(\mathcal{H}^{a,b})$

Key Result

Proposition 3.1

he Let $d$ be a superderivation of degree $0$ on the generalized quaternion algebra $\mathcal{H}^{a,b}$. Then the matrix representation $M$ of $d$ is as follows: where $\lambda,\mu,\nu\in \mathcal{R}$. Here the action of $M$ corresponds to multiplying the matrix by a column on the right.

Theorems & Definitions (18)

  • Proposition 3.1
  • Proposition 3.2
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 4.1
  • Theorem 4.1
  • proof
  • ...and 8 more