Enrico Celeghini - Università di Firenze #
Broken Symmetries #
Two objects we classify together cannot be identical as we have 
to distinguish 
between them. Thus they can only be almost equal: every symmetry 
is -by itself-
broken. Broken symmetries were, up to few decades ago, realized 
starting from
exact mathematical symmetries and introducing the breaking at 
the physical
level or in the states (as in spontaneous symmetry breaking) or 
in the
operators (as in mass formulas for hadrons).
Recently the possibility has appeared to put the breaking 
directly in 
the mathematics. This talk is devoted to discuss this new 
approach where 
deformations of Lie algebras, called quantum algebras, are 
introduced.
Quantum algebras considered as Hopf algebras are not 
deformations of Lie
algebras but deformations of Lie universal enveloping algebras, 
i.e. they are
infinite dimensional deformation of the infinite dimensional 
algebra of the
polynomials constructed on the Lie algebras generators. To close 
the play we
have thus to mimic the Cartan work to individuate the quantum 
algebra canonical
basis to be put in one-to-one correspondence with the Cartan 
basis of the Lie
algebra and successively with the physical operators of the 
unbroken symmetry.
Starting from the Lie bialgebra and using coassociativity (in 
physics, the
composed system rule) and analyticity, a self-consistent 
perturbative approach
allows us to obtain the corresponding quantum algebra in its 
canonical basis.
A consistent extension to quantum superalgebras as well as to 
quantum Poisson
algebras has been also developed.