Computing Derivations on Nilpotent Quadratic Lie Algebras

TL;DR

This paper designs different techniques to obtain any quadratic Lie algebra whose (solvable) radical ideal is nilpotent, and proposes two alternative methods, both involving the use of quotients.

Abstract

Every non-solvable and non-semisimple quadratic Lie algebra can be obtained as a double extension of a solvable quadratic Lie algebra. Thanks to a partial classification of nilpotent Lie algebras and this result, we can design different techniques to obtain any quadratic Lie algebra whose (solvable) radical ideal is nilpotent. To achieve this, we propose two alternative methods, both involving the use of quotients. In addition to their mathematical description, both approaches introduced in this paper have been computationally implemented and are publicly available to use for generating these algebras.

Figures (4)

• Figure 1: Here $N_1$ is the nilradical of $L_1$, $\mathcal{J}_1=\mathcal{J}(L_1)$ and $\mathcal{J}_1^\perp=R_1^\perp \oplus Z(L)$. For $L_1=R_1$, we have $R_1^\perp=0$.
• Figure 2: Deconstruction of $L_1(n)$.
• Figure 3: $\varphi$-skew derivations of $\mathfrak{n}_{2,5}/I_1$ computed in Wolfram Mathematica using our library.
• Figure 4: $\varphi$-skew derivations of $\mathfrak{n}_{2,5}/I_1$ computed in Wolfram Mathematica using our library.

Theorems & Definitions (12)

• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Example 3.1
• Example 3.2
• Theorem 4.1
• Remark 5.1
• Theorem 5.2
• Example 5.1