Représentation adjointe

Résumé

En mathématiques, il existe deux notions de représentations adjointes :

la représentation adjointe d'un groupe de Lie sur son algèbre de Lie,
la représentation adjointe d'une algèbre de Lie sur elle-même. Alors que la première est une représentation de groupe, la seconde est une représentation d'algèbre.

Définition Soient :

G, un groupe de Lie ;
e\in G, l'élément identité de G ;
\mathfrak{g}:=T_e G, l'algèbre de Lie de G ; *\iota:G\to\mathrm{Aut}(G);g\mapsto \iota_g l'automorphisme intérieur de G sur lui-même, donné par \iota_{g_1}(g_2)=g_1g_2g_1^{-1}.

Définition : La représentation adjointe du groupe de Lie G sur son algèbre de Lie \mathfrak{g} est : :\mathrm{Ad} : G\to\mathrm{Aut}(\mathfrak{g}) ; g\mapsto \mathrm{Ad}g := \left((\iota_g)*|_e:\mathfrak{g}\to\mathfrak{g}\right). Remarques : *la représentation adjointe

Source officielle

https://fr.wikipedia.org/wiki/Repr%C3%A9sentation_adjointe

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The possibility to explain the baryon asymmetry in the Universe through the leptogenesis mechanism in the context of Adjoint SU(5) is investigated. In this model the neutrino masses are generated through the Type I and Type III seesaw mechanisms, and the field responsible for the Type III seesaw, called rho_3, generates the B-L asymmetry needed to satisfy the observed value of the baryon asymmetry in the Universe. We find that the CP asymmetry originates only from the vertex correction, since the self-energy contribution is not present. When neutrino masses have a normal hierarchy, successful leptogenesis is possible for 10^{11} GeV < M_{\rho_3}^{NH} < 4 10^{14} GeV. When the neutrino hierarchy is inverted, the allowed mass range changes to 2 10^{11} GeV < M_{\rho_3}^{IH} < 5 10^{11} GeV. These constraints make possible to rule out a large part of the parameter space in the theory which was allowed by the unification of gauge interactions and the constraints coming from proton decay.

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On the Role of Low-Energy CP Violation in Leptogenesis

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The link between low-energy CP violation and leptogenesis became more accessible with the understanding of flavor effects. However, a definite well-motivated model where such a link occurs was still lacking. Adjoint SU(5) is a simple grand unified theory where neutrino masses are generated through the Type I and Type III seesaw mechanisms, and the lepton asymmetry is generated by the fermionic triplet responsible for the Type III seesaw. We focus exclusively on the case of inverted hierarchy for neutrinos, and we show that successful flavored leptogenesis in this theory strongly points towards low-energy CP violation. Moreover, since the range of allowed masses for the triplet is very restricted, we find that the discovery at the LHC of new states present in the theory, together with proton decay and unification of gauge couplings, can conspire to provide a hint in favor of leptogenesis.

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