Adjoint representation

A new class of effective Lagrangians have been constructed to show how the information about the center group symmetry is efficiently transferred to the actual physical states of the theory [12-15] and will be reviewed in detail elsewhere. Via these Lagrangians we were also able to have a deeper understanding of the relation between chiral restoration and deconfinement [15] for quarks in the fundamental and in the adjoint representation of the gauge group. 

Action of an affine group scheme 21 Adjoint representation of G 100 Affine algebraic group 29 Affine group scheme 5 Algebra 3... 

Next, in the Lie algebra so(3), we consider the (co)adjoint representation (action) of the group SO (3). It is readily seen that the orbits of this action are standard two-dimensional spheres centered at the origin. 

Claim 4.1.1 A field t described by the Euler equations of motion of a rigid body fixed in the centre of mass is tangent to the orbits 0 of the adjoint representation in the Lie algebra so(3) and is Hamiltonian on these orbits (which are homeomorphic to the spheres S ). 

Theorem 4.1.5 (Trofimov for particular cases, Brailov for the GENERAL CASE). Let G = if 0 pK be an extension of a compact Lie algebra K by means of the adjoint representation of K on an Abelian ideal if, that is, p = adK K K, Then on each orbit of general position in G there always exists a maximal linear commutative algebra of polynomials. Its generators are... 

Nous appellerons Lie(G/s) muni de cette structure l "algebre de Lie11 de G sur S (les guillemets etant justifies par le fait que Lie(G/S) n est pas a strictement parler une Og-algfebre de Lie). A toute representation lindaire de G dans un bon Og-module F est associ e une representation de son "algfebre de Lie". En particulier, h la representation adjointe de G est associee la representation adjointe de son "algfebre de Lie". 

Ceci s applique en particulier au cas oh on prend pour p la representation adjointe de G. A tout sous-module I de Lie(G/s) on associe done deux sous-groupes de G, son centralisateur et son normalisateur,dont les alghbres de Lie sont respectivement le centralisateur et le normalisateur... 

In a truncated coupled cluster approach, tlie two vectors are not connected by the adjoint operation but without truncations a representation of the exact state situation is retrieved and one state is the adjoint of the other. The generalized Hellmann-Feynman theorem is proven to hold... 

The HS binary product has the property that the Liouvillian superoperator L is self-adjoint according to (2.4) in this representation, and, for this reason, the HS binary product is the natural reference product in the Liouvillian formalism. 

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