Mathematical structures (extern link Wikipedia – GB)

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Mathematical structures

 

Motivations

This chapter finds its motivations in physics through at least two topics:

·       The presence, in E(3, C), of E. Cartan’s spinors at the Euclidean limit of the theory [ISBN_102-8]; this fact sheds a new light on our three-dimensional every-day world.

·       Non-trivial decomposition of the angular momentum and the Bowen-York initial data. See, for the documentation, e.g.: [3 + 1 Formalism and Basis of Numerical Relativity; § 8.2.6, pp. 136-139; arXiv: gr-qc/0703035v1, 06 March 2007] and, for the consequences of the theory of the (E) question, my document: “Bowen-York revisited”, in the French language [ISBN_113-4}.

 

 

In this chapter, all documents focus attention on mathematical structures related to the existence of deformed tensor (resp. deformed Lie) products on a vector space E(D, C); D is the natural non-vanishing dimension of that space whilst C represents the set of all complex numbers.

 

The following items will be examined:

·       Lie Algebra on V(D, A) = {E(D, C), ÄA} where A represents a cube (an element in C3).

·       Lie Algebra on W(D, ¡) = {E(D, C), […, …]¡} where ¡ represents an anti-symmetric cube (¡abc + ¡bac = 0); actually only in the French language [ISBN_136-3].

·       Involution on V(D, A).

·       Involution on W(D, ¡), in peculiar on W(3, [A]): [ISBN_116-5]; discover the main ideas.

 

 

© Thierry PERIAT, 10 May 2019.

Involution