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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 $L$,
Quote:
 Let $I$ be an ideal of $L$ and $H$ a subalgebra of $L$. Then $(I+H)/I$ is isomorphic to $H/(I\cap H)$
What I find strange is that according to my logic, $(I+H)/I$ is equal to $H/I$, since $I$ is an ideal. So why bother calling it $(I+H)/I$?

 February 1st, 2014, 04:31 AM #2 Senior Member   Joined: Mar 2012 Posts: 294 Thanks: 88 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)
 February 2nd, 2014, 01:45 AM #3 Newbie   Joined: Oct 2012 Posts: 5 Thanks: 0 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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