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Math 690A: Maximal Toral Subalgebras & Root Strings in Linear Lie Algebras, Assignments of Algebra

Iowa State University (ISU)Algebra

This problem set consists of two questions related to classical linear lie algebras. The first question asks to prove that the set of diagonal matrices in a lie algebra l of type aℓ is a maximal toral subalgebra of dimension ℓ. The second question involves calculating the root strings and cartan integers for the special linear lie algebra sl(n, f), and proving that all cartan integers for roots other than ±α are 0 or ±1.

Typology: Assignments

Pre 2010

Uploaded on 09/02/2009

koofers-user-dlz
koofers-user-dlz 🇺🇸

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Math 690A
Problem Set 7
Due: March 1, 2007
1. If Lis a classical linear Lie algebra of type A`, prove that the set of all diagonal
matrices in Lis a maximal toral subalgebra of dimension `.
2. For L=sl(n, F ), calculate explicitly the root strings and Cartan integers. In partic-
ular, prove that all Cartan integers β(hα) with β6=±α∈sl(n,F ) are 0,±1.
1

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Math 690A

Problem Set 7

Due: March 1, 2007

  1. If L is a classical linear Lie algebra of type A, prove that the set of all diagonal matrices in L is a maximal toral subalgebra of dimension.
  2. For L = sl(n, F ), calculate explicitly the root strings and Cartan integers. In partic- ular, prove that all Cartan integers β(hα) with β 6 = ±α ∈ sl(n, F ) are 0, ±1.