Timeline for $Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 23, 2015 at 17:41 | comment | added | grghxy | The unique automorphism of the Clifford algebra ${\rm{C}}(V,q)$ that extends negation on the underlying vector space $V$ of the quadratic space $(V,q)$ also swaps the two associated Clifford norms $\nu_q^{+}, \nu_q^{-}: {\rm{GPin}}(q) \rightarrow {\rm{GL}}_1$. Hence, it swaps their respective kernels ${\rm{Pin}}^{\pm}(q)$. (I am assuming these are the Pin groups you have in mind.) This applies in any dimension. | |
| Jul 23, 2015 at 16:13 | comment | added | Theo Johnson-Freyd | The isomorphism Pin^+(4k) = Pin^-(4k) is documented in "Analysis, Manifolds and Physics. Part II" by Y. Choquet-Bruhat and C. De Witt-Morette (Elsevier, 2000). Is this what you're looking for? Note that it is not an isomorphism of groups over O(4k). (The isomorphism, as you point out, covers an interesting outer automorphism of O(4k).) | |
| Jul 23, 2015 at 15:19 | history | asked | user43326 | CC BY-SA 3.0 |