Abstract
Let V be a vector space of finite dimension over a field K, and \(V^*\) the dual space; there is a canonical quadratic form on \(V\oplus V^*\), and there is a well known group morphism from \(\mathrm {GL}(V)\) into \(\mathrm {O}(V\oplus V^{*})\). This morphism can be factored through the Clifford-Lipschitz group \(\mathrm {GLip}(V \oplus V^{*})\) that lies over \(\mathrm {O}(V\oplus V^*)\); but when K is the field \(\mathbb {R}\) or \(\mathbb {C}\), it is not possible to factor it through the smaller group \(\mathrm {Spin}(V\oplus V^*)\); this follows from a theorem published by H. Bass in 1974. It is worth recalling this theorem, and comparing it with an article published by Doran, Hestenes, Sommen and Van Acker in 1993, which asserts that every Lie group can be embedded in a spinorial group. The falseness of this assertion is explained at the end of this article.
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Communicated by Rafał Abłamowicz
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Helmstetter, J. About a Theorem of Hyman Bass and some other Topics. Adv. Appl. Clifford Algebras 28, 47 (2018). https://doi.org/10.1007/s00006-018-0863-x
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DOI: https://doi.org/10.1007/s00006-018-0863-x