Consider the Clifford algebra \(C_2\) associated to the standard inner product on \(\mathbb{R}^2\). Let \(e_1\), \(e_2\) be the standard basis on \(\mathbb{R}^2\). Then\[1,e_1,e_2,e_1e_2\]is a basis of \(C_2\) where the product is given by\[e_i^2=1,~e_1e_2=-e_2e_2\]A minimal left ideal \(I\) of \(C_2\) is spanned by\[1+e_1,~e_2-e_1e_2\]
In particular, \(I\) is an irreducible module over \(C_2\). If \(X\in I\), is it correct to call \(X\) a spinor?
Diffgeometer1 /
npub14m…ctm7l
2025-01-12 17:21:48
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npub14mnqckz48cqng6rjfnvywwcy73u9ewuff8khnfk0vs7zege096fscctm7lPublished at
2025-01-12 17:21:48Event JSON
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"content": "Consider the Clifford algebra \\(C_2\\) associated to the standard inner product on \\(\\mathbb{R}^2\\). Let \\(e_1\\), \\(e_2\\) be the standard basis on \\(\\mathbb{R}^2\\). Then\\[1,e_1,e_2,e_1e_2\\]is a basis of \\(C_2\\) where the product is given by\\[e_i^2=1,~e_1e_2=-e_2e_2\\]A minimal left ideal \\(I\\) of \\(C_2\\) is spanned by\\[1+e_1,~e_2-e_1e_2\\]\nIn particular, \\(I\\) is an irreducible module over \\(C_2\\). If \\(X\\in I\\), is it correct to call \\(X\\) a spinor?",
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