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# Generalized Root Graded Lie Algebras

thesis

posted on 2018-08-23, 09:37 authored by Hogar M. YaseenLet g be a non-zero finite-dimensional split semisimple Lie algebra with root system
Δ. Let Γ be a finite set of integral weights of g containing Δ and {0}. We say that a Lie
algebra L over F is generalized root graded, or more exactly (Γ,g)-graded, if L contains
a semisimple subalgebra isomorphic to g, the g-module L is the direct sum of its weight
subspaces Lα (α ∈ Γ) and L is generated by all Lα with α ̸= 0 as a Lie algebra. If g is the
split simple Lie algebra and Γ = Δ∪{0} then L is said to be root-graded. Let g∼=
sln and
Θn = {0,±εi±ε j,±εi,±2εi | 1 ≤ i ̸= j ≤ n}
where {ε1, . . . , εn} is the set of weights of the natural sln-module. Then a Lie algebra
L is (Θn,g)-graded if and only if L is generated by g as an ideal and the g-module L
decomposes into copies of the adjoint module, the natural module V, its symmetric and
exterior squares S2V and ∧2V, their duals and the one dimensional trivial g-module.
In this thesis we study properties of generalized root graded Lie algebras and focus
our attention on (Θn, sln)-graded Lie algebras. We describe the multiplicative structures
and the coordinate algebras of (Θn, sln)-graded Lie algebras, classify these Lie algebras
and determine their central extensions.

## History

## Supervisor(s)

Baranov, Alexander## Date of award

2018-06-22## Author affiliation

Department of Mathematics## Awarding institution

University of Leicester## Qualification level

- Doctoral

## Qualification name

- PhD

## Language

en## Administrator link

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