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I am self-studying Lie algebra for the Lorentz group, and I would like to double-check if my understandings are correct, cause on many physics books the most of the time is not clearly stated if he is referring the Lie algebra or its complexification

Is the following chains of Lie algebra isomorphisms correct? (the $_\Bbb C$ stay for the complexified algebra):

$ \mathfrak su(2)_\Bbb C \cong \mathfrak sl(2, \Bbb C) $

$\mathfrak so(1,3)_\Bbb C \cong \mathfrak su(2)_\Bbb C \oplus \mathfrak su(2)_\Bbb C \cong \mathfrak sl(2, \Bbb C) \oplus \mathfrak sl(2, \Bbb C) $

$\mathfrak sl(2, \Bbb C) \cong \mathfrak sl(2, \Bbb C)_\Bbb C$

Andrea
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    That last one is incorrect. If you are viewing $\mathfrak{sl}(2,\mathbb{C})$ as a real Lie algebra then its complexification is $\mathfrak{sl}(2,\mathbb{C}) \oplus \mathfrak{sl}(2,\mathbb{C})$ (You can think of the two copies as complex conjugate to one another). – Callum Oct 09 '21 at 12:22
  • Thanks! I'am also confused by the use of ⊕ symbol, as far as I know direct sum of vector space is defined when V1∩V2=∅, since LAs are vector spaces I don't understand the meaning of sl(2,C)⊕sl(2,C). What am I missing or thinking wrong? – Andrea Oct 09 '21 at 12:55
  • Here I am using it to mean direct sum of Lie algebras. This means they are the direct sum of vector spaces and they have Lie bracket given by the Lie bracket on each one independently and the bracket of any element from one with any element from the other is 0. (You will see this notation abused somewhat but it should be clear in context). Note we think of the two terms as separate spaces (this is sometimes referred to as the exterior direct product) – Callum Oct 09 '21 at 13:04
  • as far as I know the direct sum of two vector spaces V1 and V2 requires V1∩V2=∅, so it seems we can't use direct sum when the two vector spaces are the same – Andrea Oct 09 '21 at 13:17

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