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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 1st 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexmoved the discussion of the Lie algebra $\mathfrak{su}(2)$ out of the middle of "Properties -- General" into its own subsection "Properties -- Lie algebra". Will copy this also over to _[[su(2)]]_ <a href="https://ncatlab.org/nlab/revision/diff/SU%282%29/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/SU%282%29/18">v18</a>, <a href="https://ncatlab.org/nlab/show/SU%282%29">current</a>

   moved the discussion of the Lie algebra $\mathfrak{su}(2)$ out of the middle of “Properties – General” into its own subsection “Properties – Lie algebra”.

   Will copy this also over to su(2)

   diff, v18, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 1st 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, strange. The above edit comment was for _[[SU(2)]]_ (capital)

   Hm, strange. The above edit comment was for SU(2) (capital)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2020
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   Author: Urs
   Format: MarkdownItexJust something elementary: unless I am mixed up, the wedge square of the 4d irrep of $\mathrm{SU}(2)$ is the sum of the 3d irrep and the 3d trivial rep: $$ \mathbf{4}\wedge \mathbf{4} \;\simeq\; 3 \cdot \mathbf{1} \;\oplus\; 1 \cdot \mathbf{3} \;\;\;\in\; \mathrm{RO}(\mathrm{Sp}(1)) $$ (see [[Sandbox]]) What would be a classical fact/text to refer to for this? <a href="https://ncatlab.org/nlab/revision/diff/SU%282%29/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/SU%282%29/19">v19</a>, <a href="https://ncatlab.org/nlab/show/SU%282%29">current</a>

   Just something elementary:

   unless I am mixed up, the wedge square of the 4d irrep of $\mathrm{SU}(2)$ is the sum of the 3d irrep and the 3d trivial rep:

   $$\mathbf{4}\wedge \mathbf{4} \;\simeq\; 3 \cdot \mathbf{1} \;\oplus\; 1 \cdot \mathbf{3} \;\;\;\in\; \mathrm{RO}(\mathrm{Sp}(1))$$

   (see Sandbox) What would be a classical fact/text to refer to for this?

   diff, v19, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 13th 2020
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   Author: Urs
   Format: MarkdownItexin any case, I have added the statement and proof of this elementary fact at _[[SU(2)]]_ [here](https://ncatlab.org/nlab/show/SU(2)#4Wedge4SpSp1Is3TimesTrivialPlusThree) (just to remark that the automatic announcement mechanism gets confused with the capitalization: all edits to [[SU(2)]] seem to get announced here in the thread on [[su(2)]])

   in any case, I have added the statement and proof of this elementary fact at SU(2) here

   (just to remark that the automatic announcement mechanism gets confused with the capitalization: all edits to SU(2) seem to get announced here in the thread on su(2))

5. • CommentRowNumber5.
   • CommentAuthorRichard Williamson
   • CommentTimeMar 14th 2020
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   Author: Richard Williamson
   Format: MarkdownItexThanks for raising, I have now [fixed this](https://github.com/ncatlab/nlab/commit/395af93c9992625f2574fa63709ba5b8618ffb5e). Just a reminder that I may miss this kind of thing unless it is flagged in an obvious place for me to see it.

   Thanks for raising, I have now fixed this. Just a reminder that I may miss this kind of thing unless it is flagged in an obvious place for me to see it.

6. • CommentRowNumber6.
   • CommentAuthorStableHolonomy
   • CommentTimeOct 17th 2024
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   Author: StableHolonomy
   Format: TextIs there a Pontrjagin duality for SU(2) (unit ball in the quaternions), much like how S¹ is the unit ball in the complex numbers? There is a Haar integral for SU(2), and many of the same proofs appear to carry over. But its also possible that [[G,SU(2)],SU(2)] is better supposed to be isomorphic to (G ⋊ ℤ/2ℤ) ⋊ ℤ/2ℤ

   Is there a Pontrjagin duality for SU(2) (unit ball in the quaternions), much like how S¹ is the unit ball in the complex numbers?
   
   There is a Haar integral for SU(2), and many of the same proofs appear to carry over. But its also possible that [[G,SU(2)],SU(2)] is better supposed to be isomorphic to (G ⋊ ℤ/2ℤ) ⋊ ℤ/2ℤ
7. • CommentRowNumber7.
   • CommentAuthorStableHolonomy
   • CommentTimeOct 17th 2024
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   Author: StableHolonomy
   Format: TextMaybe the group starts life with an action of ℤ/2ℤ.

   Maybe the group starts life with an action of ℤ/2ℤ.