Let n e N and define (On h := 0, 0n ) |x € M„(C)}. (i) Show that h is an n2-dimensional abelian Lie subalgebra of sl(2n). (ii) Show that h is not a Cartan subalgebra of sl(2n). This illustrates that the dimension of an abelian Lie subalgebra (here dim h = n²) can be larger than the rank of the ambient Lie algebra (here rank(sl(2n)) = 2n – 1), as long as the subalgebra is not a Cartan subalgebra. -

Let n e N and define (On h := 0, 0n ) |x € M„(C)}. (i) Show that h is an n2-dimensional abelian Lie subalgebra of sl(2n). (ii) Show that h is not a Cartan subalgebra of sl(2n). This illustrates that the dimension of an abelian Lie subalgebra (here dim h = n²) can be larger than the rank of the ambient Lie algebra (here rank(sl(2n)) = 2n – 1), as long as the subalgebra is not a Cartan subalgebra. -

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 30E: Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G...

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