Let n e N and define h := { o |x € M„(C)}. 0n 0n (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 h := { o |x € M„(C)}. 0n 0n (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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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