We generalize to the case of superalgebras several properties of simple Lie algebras involving the use of Dynkin diagrams. If to a simple Lie algebra can be associated one Dynkin diagram, it is a finite set of nonequivalent ones which can be constructed for a basic superalgebra (or B.S.A.). The knowledge of these diagrams, which can be obtained for each B.S.A. in a systematic way, allows us to deduce the regular subsuperalgebras of a B.S.A. The symmetries of the Dynkin diagrams are related to outer automorphisms of B.S.A. and lead to some singular subsuperalgebras. Finally we consider the extended Dynkin diagrams in order to classify the affine B.S.A. and use their symmetries to construct the twisted basic superalgebras.
The quantum enveloping algebra U_q(sl(2) \oplus sl(2)) in the limit q \to 0
is proposed as a symmetry algebra for the genetic code. In this approach the
triplets of nucleotids or codons in the DNA chain are classified in crystal
bases, tensor product of U_{q \to 0}(sl(2) \oplus sl(2)) representations. Such
a construction might be compared to the baryon classification from quark
building blocks in elementary particles physics, one of the main differences
standing in the property of a crystal base to provide a natural order in the
state constituents, this order being crucial in the codon. Then an operator
ensuring the correspondence codon/amino-acid can be constructed out of the
above algebra. It will be called the reading operator, and be such that two
codons relative to the same (resp. different) amino-acid(s) acquire the same
(resp. different) eigenvalue(s).Comment: LaTeX-2e document, package amsfonts, 11 page
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