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Spin and Division Algebras

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The concept of a division algebra generalizes the real and complex numbers by requiring a multiplication with suitable properties to be defined on a normed vector space. There are actually just two new possibilities in higher dimensions, known as the quaternions (dimension 4) and octonions (dimension 8). Spin, on the other hand, refers to a particular way of realizing rotational transformations in 3 or higher dimensions. The distinctive mathematical properties which emerge are also of great importance in physics, since the description of many elementary particles (quarks, electrons, neutrinos) relies on spin representations. There are some fascinating inter-relationships between spin and division algebras, leading ultimately to concepts such as triality, supersymmetry and exceptional Lie groups. My aim in this talk will be to give an overview of some of these ideas.

This talk is part of the The Archimedeans (CU Mathematical Society) series.

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