• April 21, 2017
• Symmetry And Group By Jean-Marc Levy-Leblond

A bankruptcy dedicated to Galilei staff, its representations and purposes to physics.

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Additional resources for Galilei Group and Galilean Invariance

Example text

The first equation in Lamina 3 just tells us that ~ is a map of R-algebras. All the maps occuring in the lest two equations of the lemma are homomorphism of algebras preserving identities. In each case it then suffices to verify that the images of the generators di of the algebra E(LF) coincide, and this follows from the explicit description given earlier on. PROOF o_~fTheorem i (D and w Prop. 4). (i) is just Lamina 2. (iii) follows from the fact that F ~-~ LF is a functor, end from (ii). For (ii), we recall (cf.

In the previously introduced notation for the indeterminates of R2n , h(X',X") Wn is then given by = h(X,X). Let ~n : Rn § R be the augmentation, ~n : R § R n the ring embeddirg. }[e then have, on identifying U2n = U n Q RUn, (~iting ~ for ~ R ) - \$5- PRpPOSITIO 3 The o ~n : Un § U2n = U n @ RUn , cn : R § n P de f~De on U n the structure of a_ coel~ebr~. isomorphism ' ~s rise t_~oa_ bijection Eom CRn. ) I~,00F omCo g( ,Un). For the first assertion we only have to show that ~n' en and ~n enter into commutative diagrams dual to those postulated for 9~.

We define T(Rn) to be the submodule of those u ~ U n for vhich = 0 = . T(R n) is thus the submodule generated by the A i. The next proposition gives an ~n~er characterisation of T(R n) in terms of the coalgebra structure of U n. m0POSi~O~, ~ , ai,~ u ~. u n ~ e fo~o~ st ~ t ~ n t s equivalent (ii) (iii) (Recall that ~*(u) = u O r + r ~u, = c(f) + r , r = 60 is always the identity in amy bialgebra PF structure of Un). - 43 - PROOF (i) => (ii): By Prop. 4 (ii) holds for u = Ai, hence by W R-linearity of ~ for all u E T(Rn).