# On nilpotent generators of the symplectic Lie algebra

## TL;DR

This paper proves that in the symplectic Lie algebra over an algebraically closed field of characteristic zero, any nonzero nilpotent element can be paired with another nilpotent element to generate the entire algebra.

## Contribution

It establishes that every nonzero nilpotent element in the symplectic Lie algebra can be combined with a suitable nilpotent element to generate the whole algebra, a new structural insight.

## Key findings

- Any nonzero nilpotent element can generate the algebra with another nilpotent element.
- The result holds over algebraically closed fields of characteristic zero.
- Provides a method to find such generating pairs.

## Abstract

Let $\mathfrak{sp}_{2n}(\mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element $X \in \mathfrak{sp}_{2n}(\mathbb {K})$ there exists a nilpotent element $Y \in \mathfrak{sp}_{2n}(\mathbb {K})$ such that $X$ and $Y$ generate $\mathfrak{sp}_{2n}(\mathbb {K})$.