It has been proposed [1] that the degeneracy of the genetic code, i.e., the phenomenon that different codons (base triplets) of DNA are transcribed into the same amino acid, may be interpreted as the result of a symmetry breaking process. In ref.[1] this picture was developed in the framework of simple Lie algebras. Here, we explore the possibility of explaining the degeneracy of the genetic code using basic classical Lie superalgebras, whose representation theory is sufficiently well understood, at least as far as typical representations are concerned. In the present paper, we give the complete list of all typical codon representations (typical 64-dimensional irreducible representations), whereas in the second part, we shall present the corresponding branching rules and discuss which of them reproduce the multiplet structure of the genetic code.

Lie superalgebras and the multiplet structure of the genetic code. I. Codon representations Continuing our attempt to explain the degeneracy of the genetic code using basic classical Lie superalgebras, we present the branching schemes for the typical codon representations ͑typical 64-dimensional irreducible representations͒ of basic classical Lie superalgebras and find three schemes that do reproduce the degeneracies of the standard code, based on the orthosymplectic algebra osp͑5͉2͒ and differing only in details of the symmetry breaking pattern during the last step.

We discuss quantum algebraic structures of the systems of electrons or quasiparticles on a sphere of which center a magnetic monople is located on. We verify that the deformation parameter is related to the filling ratio of the particles in each case.1

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A nonperturbative symmetry that relates the weak and strong coupling regimes of a quantum theory. In a gauge theory the inversion of the coupling is accompanied by the interchange of the electric and magnetic degrees of freedom [1]. When a h-term is included in the Lagrangian there is an additional symmetry under h ! h þ 2p, and the theory is then invariant under s ! À1=s and s ! s þ 1, withand g the coupling constant. These two transformations generate the SLð2; Z Þ group, or S-duality group [2]. Examples in which it is conjectured to be a symmetry: Four dimensional N¼4 supersymmetric Yang Mills theory, where it is referred as Montonen Olive duality [3]. In string theory: symmetry of the ten dimensional Type IIB superstring theory [4], it relates the Type I and heterotic SO(32) theories [5,6,7], in four dimensions it is a symmetry of the heterotic compactified on a 6-torus [8].