Seite - 64 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 part is a coadjoint representationof G, that is the contragradientof theadjoint representation. It associates to eachg∈Gthe linear isomorphismAd∗g∈GL(g∗), satisfying, for each: ξ∈ g∗ andX∈ g : 〈 Ad∗g(ξ),X 〉 = 〈 ξ,Adg−1(X) 〉 . Then, the fundamental equationsofLiegroup thermodynamicsaregivenby theactionof thegroup: • ActionofLiegrouponLiealgebra: β→Adg(β) (37) • Transformationof characteristic functionafter actionofLiegroup: Φ→Φ− 〈 θ ( g−1 ) ,β 〉 (38) • Invarianceof entropywith respect toactionofLiegroup: s→ s (39) • ActionofLiegroupongeometricheat, elementofdualLie algebra: Q→ a(g,Q)=Ad∗g(Q)+θ(g) (40) SouriauequationsofLiegroupthermodynamicsaresummarizedinthefollowingFigures5and6: Figure5.GlobalSouriauschemeofLiegroupthermodynamics. 64