Marmo G. A Manifold view point of Lie Algebras (abstract)

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Speaker: Giuseppe Marmo

Title: A Manifold view point of Lie Algebras

Abstract:
Lie algebra brackets play a prominent role in the description of evolution (equations of motion) for any physical system, be it classical or quantum. On the other hand, the advent of general relativity has called for a description of physical systems in a coordinate independent manner. Special relativity has introduced the need for a composition law for velocities whose expression [math]\displaystyle{ (v+w)/[1+(vw/c^2)] }[/math] is alternative to the usual Galilean one [math]\displaystyle{ v+w }[/math], which appears to be a "contraction" of previous one when [math]\displaystyle{ c }[/math], the speed of light, goes to infinity. The composition law of special relativity is not Archimedean and is not associative in usual space-time (1+3). However in 1+1 space-time it defines an alternative "linear structure".

What we learn from these observations is that from the point of view of Physics it would be convenient to have a "tensorial" presentation of Lie algebras where not only the binary, bilinear product is represented by a tensor but the linear structure itself be represented by a tensor field. These aspects have been tackled in some previous papers by our research group and, as we shall show, have a strong relation with some of latest published papers by Alexander Vinogradov on the classification of finite dimensional real Lie algebras [Particle-like structure of Lie algebras].


Event: Diffieties, Cohomological Physics, and Other Animals, 13-17 December 2021, Moscow.
Alexandre Vinogradov Memorial Conference.