| We give two realizations respectively for representations of afne Lie algebra A1（1）1and the relevant Z-algebra in terms of vertex operators with level k. Afne Lie algebrais a very important kind of infinite symmetry in quantum field theory. And the studyof the representation of afne Lie algebras has been perhaps one of most popular issuein mathematical physics theory. Difers from the finite dimensional case, the algebragenerators of infinite dimensional Lie algebra must use the operators in infinite space torealize.We get the module of infinite afne Lie algebra at integer level generally by the vertexoperator. But the previous method is incompatible with the case of negative integer level. Dunbar Jonathan, Jing N, Misra Kailash recently[10]give a new method to constructmodule of （?）sl2（（C））A1（1）A1at critical level using the generalized Cliford algebra. Consid-ering all negative integer level k, we promote the their work of critical level. However,operator product or even the representation space we constructed need certain revision,and that the parafermions theory has to be introduced into the theory.The first part of this article gives an overview of the theoretical significance of therepresentations of afne Lie algebra with concrete realizations with respect to a certainvertex operators, and of the introduction of the concept of parafermions in modern the-oretical physics application.The second part reviews some recent results done by thepaper[10], and briefly explains the physicists approach to parafermion field with respectto each form of current algebra,operator product expansion and so on, also studys theexchange relationship for parafermions and does some modifications as need. The thirdpart gives the main results.A parafermion representation of A1（1）is obtained by construct-ing operators and some space. Further, we do the similar work for Z-algebra. Finally wepointed out that both representations are irreducible, the proof remains to be solved. |
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