Finite presentability of universal central extensions of ${\mathfrak{sl}_n}$, II

TL;DR

This paper explores the conditions under which universal central extensions of special linear Lie algebras over algebras are finitely presentable, linking Jordan algebra properties to Lie algebra structures and resolving an open question.

Contribution

It establishes a connection between finite presentability of Jordan algebras and their associated Lie algebras, completing the classification for universal central extensions of ${rak{sl}_n(A)}$.

Findings

Finite presentability of Jordan algebras implies that of their Tits-Kantor-Koecher algebras.

The paper completes the classification of finite presentability for universal central extensions of ${rak{sl}_n(A)}$.

It positively answers a question by Shestakov-Zelmanov.

Abstract

In this note we connect finite presentability of a Jordan algebra to finite presentability of its Tits-Kantor-Koecher algebra. Through this we complete our discussion of finite presentability of universal central extensions of ${\mathfrak{sl}_n(A)}$, $A$ a $k$-algebra, initiated in our previous paper, and answer a question raised by Shestakov-Zelmanov in the positive.