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January 31st, 2014, 10:02 AM   #1
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Factor space of Lie Algebras

In Roger Carter's Lie Algebras of Finite and Affine Type, Proposition 1.7(iii) states that for a Lie algebra ,
Quote:
Let be an ideal of and a subalgebra of . Then is isomorphic to
What I find strange is that according to my logic, is equal to , since is an ideal. So why bother calling it ?
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February 1st, 2014, 04:31 AM   #2
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Re: Factor space of Lie Algebras

Isn't I an ideal of L? There seems to be to be no guarantee than I is contained within H.

Half the battle in such a statement seems to be to be in showing I is an ideal of the subalgebra I + H (if I recall correctly, ideals are subalgebras, but not vice versa, necessarily), and that I?H is an ideal of H.

The isomorphism in question is probably intended to be, for x in I, y in H:

(x + y) + I ---> y + (I?H)
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February 2nd, 2014, 01:45 AM   #3
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Re: Factor space of Lie Algebras

Oh, of course, H/I does not make sense unless I is a subset of H. That is the reason why we have to add I to H first. Thanks for your answer!
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