Advanced Topics LIE ALGEBRAS (Date Due: 14th August) AS

Advanced Topics LIE ALGEBRAS (Date Due: 14th August) ASSIGNMENT 3 Question 1. Prove that fadx j x 2 Lg the set of all inner derivations of the Lie algebra L is an ideal in Der(L). Question 2. Show that the derived algebra of gl(n; F) is sl(n; F). Question 3. Determine the derived series and lower central series for the Lie algebra t(n; F) of upper-triangular matrices. Question 4. The subalgebra K of the Lie algebra L is called subinvariant in L if and only if there is a chain of subalgebras K =: Ls Ls??1 L2 L1 := L such that Li is an ideal of Li??1 for i = 1; : : : s. Prove that every subalgebra of a nilpotent Lie algebra is subinvariant. 1
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