A Lie algebra L is said to be reductive if Z(L) = Rad(L). (For parts (b), (c), assume chark = 0 andk algebraically closed.) (a) Prove that a reductive Lie algebra is a direct sum of an abelian Lie algebra and a semisimple Lie algebra. (b) Assume that L has a faithful irreducible finite dimensional representation. Prove that L is reductive. (c) Assume that L has a faithful completely reducible finite dimensional represen- tation. Prove that L is reductive.

A Lie algebra L is said to be reductive if Z(L) = Rad(L). (For parts (b), (c), assume chark = 0 andk algebraically closed.) (a) Prove that a reductive Lie algebra is a direct sum of an abelian Lie algebra and a semisimple Lie algebra. (b) Assume that L has a faithful irreducible finite dimensional representation. Prove that L is reductive. (c) Assume that L has a faithful completely reducible finite dimensional represen- tation. Prove that L is reductive.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 27E

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