Working a bit with Clifford algebras again. The Clifford algebra Cl(p,q) is the real algebra freely generated by p square roots of 1 and q square roots of -1, all of which anticommute. For example Cl(0,1) is the complex numbers, ℂ, because you get that by throwing in a square root of -1. Cl(0,2) is the quaternions, ℍ, because you get that by throwing in two anticommuting square roots of -1. (Their product gives a third one.)
Clifford algebras obey a bunch of cool relations, but not all were listed on Wikipedia or the nLab until today... when I put them on. Here are the most important:
Cl(q+2,p) ≅ Cl(p,q) ⊗ Cl(2,0)
Cl(q,p+2) ≅ Cl(p,q) ⊗ Cl(0,2)
Cl(p+1,q+1) ≅ Cl(p,q) ⊗ Cl(1,1)
Cl(1,1) and Cl(2,0) are both isomorphic to the algebra of 2×2 real matrices, so the last isomorphism says we can get Cl(p+1,q+1) by taking 2×2 matrices with entries in Cl(p,q) - super-useful, and this was on Wikipedia and the nLab.
But the other two isomorphisms deserve their place in the sun too!
https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras#Symmetries
John Carlos Baez
npub1nf…3nqe4
2025-11-24 09:51:41 UTC
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2025-11-24 09:51:41 UTCEvent JSON
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"content": "Working a bit with Clifford algebras again. The Clifford algebra Cl(p,q) is the real algebra freely generated by p square roots of 1 and q square roots of -1, all of which anticommute. For example Cl(0,1) is the complex numbers, ℂ, because you get that by throwing in a square root of -1. Cl(0,2) is the quaternions, ℍ, because you get that by throwing in two anticommuting square roots of -1. (Their product gives a third one.) \n\nClifford algebras obey a bunch of cool relations, but not all were listed on Wikipedia or the nLab until today... when I put them on. Here are the most important:\n\nCl(q+2,p) ≅ Cl(p,q) ⊗ Cl(2,0)\n\nCl(q,p+2) ≅ Cl(p,q) ⊗ Cl(0,2)\n\nCl(p+1,q+1) ≅ Cl(p,q) ⊗ Cl(1,1)\n\nCl(1,1) and Cl(2,0) are both isomorphic to the algebra of 2×2 real matrices, so the last isomorphism says we can get Cl(p+1,q+1) by taking 2×2 matrices with entries in Cl(p,q) - super-useful, and this was on Wikipedia and the nLab. \n\nBut the other two isomorphisms deserve their place in the sun too!\n\nhttps://en.wikipedia.org/wiki/Classification_of_Clifford_algebras#Symmetries",
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