2018YaseenHMPhD.pdf (680.72 kB)
Download file

Generalized Root Graded Lie Algebras

Download (680.72 kB)
posted on 23.08.2018, 09:37 by Hogar M. Yaseen
Let 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.



Baranov, Alexander

Date of award


Author affiliation

Department of Mathematics

Awarding institution

University of Leicester

Qualification level


Qualification name




Usage metrics