Division algebras

Division algebras

A division algebra is a mathematical structure in which every nonzero element has a multiplicative inverse. Real division algebras can only exist in dimensions 1, 2, 4, and 8, but there are other examples such as ℂ, ℍ, and 𝕆. These algebras are still division algebras, but they are not alternative and they are not unital. In a division algebra, every nonzero element is invertible.From: Handbook of Linear Algebra [2006], Advanced Topics in Mathematical Analysis [2019]

Nonassociative Algebras

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Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006

Murray R. Bremner, Lucia I. Murakami, Ivan P. Shestakov

Real division algebras [EHH91, Part B]. In the previous example, taking F to be the field ℝ of real numbers and a = b = c = −1, we obtain the field ℂ of complex numbers, the associative division algebra ℍ of quaternions, and the alternative division algebra 𝕆 of octonions (also known as the Cayley numbers). Real division algebras exist only in dimensions 1, 2, 4, and 8, but there are many other examples: The algebras ℂ, ℍ, and 𝕆 with the multiplication x⋅y=x¯y¯ are still division algebras, but they are not alternative and they are not unital.

Implications of causality for quantum biology – I: topology change

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Published in Molecular Physics, 2018

D. F. Scofield, T. C. Collins

(VFL, [11,27]): Let be a division algebra, then the gauge potential 1-form A is obtained for any contractible subdomain of a QMST manifold with conserved are n × n, -valued algebraic currents, ⋆J-conservation (equivalently, the continuity equation) via the vortex field equations (VFEs, as in Table 1):

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Nonassociative Algebras

Implications of causality for quantum biology – I: topology change