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Contractions on ranks and quaternion types in Clifford algebras. (Russian. English summary) Zbl 1413.15046
Summary: In this paper we consider expressions in real and complex Clifford algebras, which we call contractions or averaging. We consider contractions of arbitrary Clifford algebra element. Each contraction is a sum of several summands with different basis elements of Clifford algebra. We consider even and odd contractions, contractions on ranks and contractions on quaternion types. We present relation between these contractions and projection operations onto fixed subspaces of Clifford algebras – even and odd subspaces, subspaces of fixed ranks and subspaces of fixed quaternion types. Using method of contractions we present solutions of system of commutator equations in Clifford algebras. The cases of commutator and anticommutator are the most important. These results can be used in the study of different field theory equations, for example, Yang-Mills equations, primitive field equation and others.

15A66 Clifford algebras, spinors
15B33 Matrices over special rings (quaternions, finite fields, etc.)
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
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