Representation of superalgebras and their quantum groups

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In the research of this year, I found a new method to construct the N=2 superconformal algebra in terms of the affine superralgebras sl (2,1)^*. This construction has a very curious and interesting aspect ; namely in this construction, odd roots with length zero play an important role, and so the similar method does not work for usual (affine) Lie algebras. Further I extended this method to construct the N=4 superconformal algebra in terms of affine superalgebras. A paper containing these results is now in preparation.I made also some investigations and observations on the structure and representations of "conformal-superalgebras" , which are new algebraic structures, recently discovered by V.G,Kac, closely related to the super-conformal algebras. They have a great advantage that their calculation is much simpler and easier than that of the usual superconformal algebras or vertex operator algebras. Their representation theory, however, is quite different from that of superconformal algebras or even from that of usual simple Lie algebras, and we observe many curious phenominia in their representations. So their representation theory has its own interests in itself, and provides a lot of new problems which should be investigated. This year I made a joint research with V.G.Kac and S.-J.cheng on the extension of irreducible representations of conformal superalgebras, from which we can see that the representation of "conformal-superalgebras" is very rich.