Central extension of graded Lie algebras

TL;DR

This thesis investigates the universal central extension of root-graded Lie algebras, revealing that such extensions are generally not root-graded and introducing measures for this deviation, especially for type A and C root systems.

Contribution

It provides a detailed analysis of the universal central extension of root-graded Lie algebras and introduces the concept of degenerate sums to quantify their deviation from root-graded structure.

Findings

Universal central extension often not root-graded

Degenerate sums measure deviation from root-graded structure

Application of Jordan algebra theory to type A and C cases

Abstract

In this thesis we describe the universal central extension of two important classes of so-called root-graded Lie algebras defined over a commutative associative unital ring $k.$ Root-graded Lie algebras are Lie algebras which are graded by the root lattice of a locally finite root system and contain enough $\mathfrak{sl}_2$-triples to separate the homogeneous spaces of the grading. Examples include the infinite rank analogs of the simple finite-dimensional complex Lie algebras. \\ In the thesis we show that in general the universal central extension of a root-graded Lie algebra $L$ is not root-graded anymore, but that we can measure quite easily how far it is away from being so, using the notion of degenerate sums, introduced by van der Kallen. We then concentrate on root-graded Lie algebras which are graded by the root systems of type $A$ with rank at least 2 and of type $C$. For them one can use the theory of Jordan algebras.