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I know that the distinction of real / complex Lie algebra.

For example,

  • The $su(2)=so(3)=sp(1)$ is a real Lie algebra.

  • The $sl(2,\mathbf{C})=so(1,3)$ is a complex Lie algebra.

Question 1: How are their representations, being real / pseudoreal/ complex representations, related to the fact whether their Lie algebras are real / complex Lie algebra or not?

  • The $su(2)$ fundamental rep is pseudoreal. It is a 2-dimensional rep of $su(2)$.

  • The $sl(2,\mathbf{C})$ fundamental rep is complex. It is a (2,0) or (0,2)-dimensional rep of $su(2)$.

Question 2: We can consider the spin group. Note that all spin groups has real Lie algebras. Such that

  • $Spin(3)=SU(2)$ has its spinor rep is pseudoreal.

  • $Spin(4)=SU(2)\times SU(2)$ has its spinor rep is pseudoreal.

  • $Spin(5)$ has its spinor rep is pseudoreal.

  • $Spin(6)$ has its spinor rep is complex.

  • $Spin(7)$ has its spinor rep is real.

  • $Spin(8)$ has its spinor rep is real.

  • $Spin(9)$ has its spinor rep is real.

  • $Spin(10)$ has its spinor rep is complex.

And there is a modular 8 Bott periodicity.

So how are the real / complex Lie algebra $\Rightarrow$ determine Real / pseudoreal/ complex representations? Maybe I have missed some important ingredients. Please spell out and clarify the conceptual relations between the twos.

Марина Marina S
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  • The (unfortunate) terminology "real/pseudoreal/complex" refers to a trichotomy of representations of real Lie groups/algebras on complex vectorspaces. I.e. all the reps you are looking at are reps of real objects on complex vectorspaces. These fall into these three subcategories. You must also mean that by "the fundamental rep of $sl(2, \mathbb C)$" i.e. you should understand and/or clarify that here $sl(2, \mathbb C)$ is viewed as a real, not as a complex Lie algebra. Also, cf. https://math.stackexchange.com/a/4026224/96384, https://math.stackexchange.com/a/2774741/96384. – Torsten Schoeneberg Aug 04 '21 at 03:25
  • thanks - but I am confused -- I thought real Lie groups/algebras on real vector spaces with real coefficients of the vectors. I thought the real Lie group and real Lie algebra is over the real number for real vector space. – Марина Marina S Aug 04 '21 at 03:58
  • I thought complex Lie groups/algebras on complex vector spaces with complex coefficients of the vectors. I thought the complex Lie group and complex Lie algebra is over the complex number for complex vector space. – Марина Marina S Aug 04 '21 at 03:58
  • Can you write an answer on it? please – Марина Marina S Aug 04 '21 at 03:59
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    Well yes, a real Lie algebra is itself a real vector space and a complex Lie algebra a complex vector space. But now we are talking about representations of such objects. And there one has another vector space $V$ on which your object acts, and this vector space can, a priori, be real or complex regardless of what your Lie algebra is. In practice, usually the $V$ is assumed to be complex. As said, the interesting trichotomy then happens if the Lie algebra is not complex, but real. -- If I find time to write an answer, I would mostly repeat myself and/or copy-paste from above links. – Torsten Schoeneberg Aug 04 '21 at 04:48
  • that helps. Why $V$ is assumed to be complex usually? – Марина Marina S Aug 04 '21 at 05:07
  • So should I say that the representation $\rho$ maps from $$ Spin(n) \mapsto SL(n,) $$ or $$ SU(n) \mapsto SL(n,) $$ also $$ SO(n) \mapsto SL(n,) $$ even though $Spin(n)$, $SU(n)$, $SO(n)$ all are real Lie groups? but map to a complex Lie group $SL(n,)$? – Марина Marina S Aug 04 '21 at 05:12
  • It will be nice to post an answer. :) – Марина Marina S Aug 04 '21 at 17:19
  • You are right in "So should I say ..."! (Maybe a priori the $\rho$ even goes to $GL(n, \mathbb C)$, but often indeed the image lies in $SL$.) – Torsten Schoeneberg Aug 04 '21 at 17:21
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    This also https://math.stackexchange.com/q/4093823/955245 – Марина Marina S Aug 04 '21 at 21:54

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