2020/03/31 by Hogir Mohammed Yaseen, Yaseen, Hogir Mohammed
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #ams
paper · pdf · doi:10.48550/arxiv.2003.14352
openalex publication_date 2020/03/31 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
Let \mathbbF be a field of characteristic zero and let \mathfrakg be a non-zero finite-dimensional split semisimple Lie algebra with root system Δ. Let Γ be a finite set of integral weights of \mathfrakg containing Δ and \0\. Following [2,10], we say that a Lie algebra L over \mathbbF is generalized root graded, or more exactly (Γ,\mathfrakg)-graded, if L contains a semisimple subalgebra isomorphic to \mathfrakg, the \mathfrakg-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. Let \mathfrakg≅ sln and Θn = \0,±εi ±εj, ±εi, ±2εi |1 ≤ i ≠ j ≤ n\ where \ε1, …, εn\ is the set of weights of the natural sln-module. In [9], we classify (Θn,sln)-graded Lie algebras for n>4. In this paper we describe the multiplicative structures and the coordinate algebras of (Θn,sln)-graded Lie algebras (n=3,4). In n=3, we assume that [V(2ω1)⊗ C,V(2ω1)⊗ C]=[V(2ω2)⊗ C',V(2ω2)⊗ C']=0 where V(ω) is the simple \mathfrakg-module of highest weight ω, C=\rm Hom_\mathfrakg(V(2ω1),L) and C'=\rm Hom_\mathfrakg(V(2ω2),L) .