Hello, I'm studying the Lorentz group and their properties... and I have some question for them..(adsbygoogle = window.adsbygoogle || []).push({});

Peskin's text(p496) said that

"we are primarily interested in Lie algebras that have finite-dimensional Hermitian representations, leading to finite-dimensional unitary representations of the corresponding Lie group.

We will also assume that the number of generators is finite.

Such Lie algebras are called compact, because these conditions imply that the Lie group is a finite-dimensional compact manifold."

From this mentions of compactness,

I've thought that Homogeneous Lorentz group is compact because the number of generators of it is just 10([itex]M_{\mu\nu},P_{\mu}[/itex] 3 rotations, 3 boosts, 4 translations)

In Arfken and many other texts, However, they said that the Homogeneous Lorentz group is non-compact, because the limit of a sequence of rapidities going to infinity is no longer an element of the an element of the group. (Arfken, p278)

From the same reason, translation is non-compact, too.

(therefore, Poincare group is non-compact.)

What's wrong with my assumption for compactness of the Lorentz group?

I know that I have some misunderstanding for compactness and I need a help to figure it out...

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# Non-compactness of Lorentz Group ?

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